The subject error appears when I refresh the page (databinding is done on page load). Stepping through the page's code reveals no errors and no page is referenced in the stack trace:
Description: An unhandled exception occurred during the execution of the current web request. Please review the stack trace for more information about the error and where it originated in the code. 
Exception Details: System.Web.HttpException: Cannot have multiple items selected in a DropDownList.
Source Error: 
An unhandled exception was generated during the execution of the current web request. Information regarding the origin and location of the exception can be identified using the exception stack trace below.
Stack Trace:
[HttpException (0x80004005): Cannot have multiple items selected in a DropDownList.]
   System.Web.UI.WebControls.DropDownList.VerifyMultiSelect() +106
   System.Web.UI.WebControls.ListControl.RenderContents(HtmlTextWriter writer) +161
   System.Web.UI.WebControls.WebControl.Render(HtmlTextWriter writer) +32
   System.Web.UI.Control.RenderControlInternal(HtmlTextWriter writer, ControlAdapter adapter) +27
   System.Web.UI.Control.RenderControl(HtmlTextWriter writer, ControlAdapter adapter) +100
   System.Web.UI.Control.RenderControl(HtmlTextWriter writer) +25
   System.Web.UI.Control.RenderChildrenInternal(HtmlTextWriter writer, ICollection children) +208
   System.Web.UI.Control.RenderChildren(HtmlTextWriter writer) +8
   System.Web.UI.WebControls.WebControl.RenderContents(HtmlTextWriter writer) +10
   System.Web.UI.WebControls.WebControl.Render(HtmlTextWriter writer) +32
   System.Web.UI.Control.RenderControlInternal(HtmlTextWriter writer, ControlAdapter adapter) +27
   System.Web.UI.Control.RenderControl(HtmlTextWriter writer, ControlAdapter adapter) +100
   System.Web.UI.Control.RenderControl(HtmlTextWriter writer) +25
   System.Web.UI.Control.RenderChildrenInternal(HtmlTextWriter writer, ICollection children) +208
   System.Web.UI.Control.RenderChildren(HtmlTextWriter writer) +8
   System.Web.UI.Control.Render(HtmlTextWriter writer) +10
   System.Web.UI.Control.RenderControlInternal(HtmlTextWriter writer, ControlAdapter adapter) +27
   System.Web.UI.Control.RenderControl(HtmlTextWriter writer, ControlAdapter adapter) +100
   System.Web.UI.Control.RenderControl(HtmlTextWriter writer) +25
   System.Web.UI.Control.RenderChildrenInternal(HtmlTextWriter writer, ICollection children) +208
   System.Web.UI.UpdatePanel.RenderChildren(HtmlTextWriter writer) +256
   System.Web.UI.UpdatePanel.Render(HtmlTextWriter writer) +37
   System.Web.UI.Control.RenderControlInternal(HtmlTextWriter writer, ControlAdapter adapter) +27
   System.Web.UI.Control.RenderControl(HtmlTextWriter writer, ControlAdapter adapter) +100
   System.Web.UI.Control.RenderControl(HtmlTextWriter writer) +25
   System.Web.UI.Control.RenderChildrenInternal(HtmlTextWriter writer, ICollection children) +208
   System.Web.UI.Control.RenderChildren(HtmlTextWriter writer) +8
   System.Web.UI.Control.Render(HtmlTextWriter writer) +10
   System.Web.UI.Control.RenderControlInternal(HtmlTextWriter writer, ControlAdapter adapter) +27
   System.Web.UI.Control.RenderControl(HtmlTextWriter writer, ControlAdapter adapter) +100
   System.Web.UI.Control.RenderControl(HtmlTextWriter writer) +25
   System.Web.UI.Control.RenderChildrenInternal(HtmlTextWriter writer, ICollection children) +208
   System.Web.UI.HtmlControls.HtmlForm.RenderChildren(HtmlTextWriter writer) +173
   System.Web.UI.HtmlControls.HtmlContainerControl.Render(HtmlTextWriter writer) +31
   System.Web.UI.HtmlControls.HtmlForm.Render(HtmlTextWriter output) +53
   System.Web.UI.Control.RenderControlInternal(HtmlTextWriter writer, ControlAdapter adapter) +27
   System.Web.UI.Control.RenderControl(HtmlTextWriter writer, ControlAdapter adapter) +100
   System.Web.UI.HtmlControls.HtmlForm.RenderControl(HtmlTextWriter writer) +40
   System.Web.UI.Control.RenderChildrenInternal(HtmlTextWriter writer, ICollection children) +208
   System.Web.UI.Control.RenderChildren(HtmlTextWriter writer) +8
   System.Web.UI.Control.Render(HtmlTextWriter writer) +10
   System.Web.UI.Control.RenderControlInternal(HtmlTextWriter writer, ControlAdapter adapter) +27
   System.Web.UI.Control.RenderControl(HtmlTextWriter writer, ControlAdapter adapter) +100
   System.Web.UI.Control.RenderControl(HtmlTextWriter writer) +25
   System.Web.UI.Control.RenderChildrenInternal(HtmlTextWriter writer, ICollection children) +208
   System.Web.UI.Control.RenderChildren(HtmlTextWriter writer) +8
   System.Web.UI.Page.Render(HtmlTextWriter writer) +29
   System.Web.UI.Control.RenderControlInternal(HtmlTextWriter writer, ControlAdapter adapter) +27
   System.Web.UI.Control.RenderControl(HtmlTextWriter writer, ControlAdapter adapter) +100
   System.Web.UI.Control.RenderControl(HtmlTextWriter writer) +25
   System.Web.UI.Page.ProcessRequestMain(Boolean includeStagesBeforeAsyncPoint, Boolean includeStagesAfterAsyncPoint) +3060
When last this happened, I had this image tag on the master page:
<img src="<%=Page.ResoveUrl("~/Images/footerLogo.png") %>" />
Modifying this to remove the <% %> (<img src="/Images/footerLogo.png" />) block fixed the error, but I don't have any such tags on the master page this time (or any page that links to the page in question). Here's my code:
Imports MySql.Data
Imports MySql.Data.MySqlClient
Partial Class maintreservation
    Inherits System.Web.UI.Page
    Protected Sub Page_Load(ByVal sender As Object, ByVal e As System.EventArgs) Handles Me.Load
        If Not IsPostBack Then
            getClients()
            getDrivers()
            getVehicles()
            pnlDriven.Visible = False
            pnlSelfDriven.Visible = False
            If Not String.IsNullOrEmpty(Session("resid")) AndAlso Session("resid") IsNot Nothing Then
                getReservation()
            End If
        End If
    End Sub
    Protected Sub getDrivers()
        ddlDriver.ClearSelection()
        Dim db As New Database
        Dim sql As String = "select driverid, drivername from drivers order by drivername"
        Dim dr As MySqlDataReader = db.GetReader(sql)
        If dr.Read Then
            ddlDriver.DataSource = dr
            ddlDriver.DataTextField = "drivername"
            ddlDriver.DataValueField = "driverid"
            ddlDriver.DataBind()
        End If
        dr.Close()
        ddlDriver.Items.Insert(0, New ListItem("Select a Driver", 0))
    End Sub
    Protected Sub getVehicles()
        ddlVehicle.ClearSelection()
        Dim db As New Database
        Dim sql As String = "select veh.vehicleid, veh.licence_plate, veh.model, vm.make from vehicles veh " & _
            "inner join vehicle_manufacturers vm on veh.make = vm.makeid order by veh.make"
        Dim dr As MySqlDataReader = db.GetReader(sql)
        While dr.Read
            ddlVehicle.Items.Add(New ListItem(dr("make") & " " & dr("model") & " " & " (" & dr("licence_plate") & ")", dr("vehicleid")))
        End While
        dr.Close()
        ddlVehicle.Items.Insert(0, New ListItem("Select a Vehicle", 0))
    End Sub
    Protected Sub getClients()
        ddlClient.ClearSelection()
        Dim db As New Database
        Dim sql As String = "select clientid, clientname, contact from clients order by clientname"
        Dim dr As MySqlDataReader = db.GetReader(sql)
        If dr.Read Then
            ddlClient.DataSource = dr
            ddlClient.DataTextField = "clientname"
            ddlClient.DataValueField = "clientid"
            ddlClient.DataBind()
        End If
        dr.Close()
        ddlClient.Items.Insert(0, New ListItem("Select an Agent", 0))
        getClientContact()
    End Sub
    Protected Sub getClientContact()
        Dim db As New Database
        Dim sql As String = "select contact, vat_no from clients where clientid = " & ddlClient.SelectedItem.Value
        Dim dr As MySqlDataReader = db.GetReader(sql)
        If dr.Read Then
            If Not IsDBNull(dr("contact")) Then
                lblClientContact.Text = dr("contact")
            Else
                lblClientContact.Text = ""
            End If
            If Not IsDBNull(dr("vat_no")) Then
                txtVatNo.Text = dr("vat_no")
            Else
                txtVatNo.Text = ""
            End If
        End If
        dr.Close()
    End Sub
    Protected Sub ddlClient_SelectedIndexChanged(ByVal sender As Object, ByVal e As System.EventArgs) Handles ddlClient.SelectedIndexChanged
        getClientContact()
    End Sub
    Protected Sub getReservation()
        Dim db As New Database
        Dim sql As String = _
            "select * from reservations res " & _
            "inner join reservationbody trip on res.reservationid = trip.reservationid " & _
            "inner join vehicles vehicle on trip.vehicleid = vehicle.vehicleid " & _
            "left outer join drivers driver on trip.driverid = driver.driverid " & _
            "where res.reservationid = " & Session("resid")
        Dim dr As MySqlDataReader = db.GetReader(sql)
        If dr.Read Then
            ' GET CLIENT
            ' -------------------------------------------------------------
            ddlClient.ClearSelection()
            ddlClient.Items.FindByValue(dr("clientid")).Selected = True
            getClientContact()
            ' GET VOUCHER
            ' -------------------------------------------------------------
            If IsDBNull("vouchernumber") Then
                txtVoucher.Text = ""
            Else
                txtVoucher.Text = dr("vouchernumber")
            End If
            ' GET DRIVER
            ' -------------------------------------------------------------
            If dr("driverid") = "0" Then
                ' Self Drive
                rblTrips.Items(1).Selected = True
                pnlDriven.Visible = False
                pnlSelfDriven.Visible = True
            Else
                ' Driven
                rblTrips.Items(0).Selected = True
                pnlSelfDriven.Visible = False
                pnlDriven.Visible = True
                ddlDriver.ClearSelection()
                ddlDriver.Items.FindByValue(dr("driverid")).Selected = True
            End If
            ' GET PASSENGER/S
            ' -------------------------------------------------------------
            If IsDBNull(dr("passengers")) Then
                txtPassengers.Text = ""
            Else
                txtPassengers.Text = dr("passengers")
            End If
            If IsDBNull(dr("passengercontact")) Then
                txtPassengerContact.Text = ""
            Else
                txtPassengerContact.Text = dr("passengercontact")
            End If
            ' GET COMMENTS
            ' -------------------------------------------------------------
            If IsDBNull(dr("comments")) Then
                txtComments.Text = ""
            Else
                txtComments.Text = dr("comments")
            End If
            ' GET VEHICLES
            ' -------------------------------------------------------------
            ddlVehicle.ClearSelection()
            ddlDriver.Items.FindByValue(dr("vehicleid")).Selected = True
            ' GET DATES AND TIMES
            ' -------------------------------------------------------------
            txtPickupDate.Text = Left(dr("res_date"), 10) ' USE DATE(trip.res_date) as res_date in the sql statement
            ucTimeIn.Time = dr("time_in")
            ucTimeOut.Time = dr("time_out")
        End If
        dr.Close()
    End Sub
    Protected Sub rblTrips_SelectedIndexChanged(ByVal sender As Object, ByVal e As System.EventArgs) Handles rblTrips.SelectedIndexChanged
        If rblTrips.SelectedItem.Value = "driven" Then
            pnlDriven.Visible = True
            pnlSelfDriven.Visible = False
        Else
            pnlSelfDriven.Visible = True
            pnlDriven.Visible = False
        End If
    End Sub
End Class
As stated, this code executes perfectly, so I'm really at a loss as to what the problem is here.
I've tried Cleaning and Rebuilding the Solution, but it still hasn't fixed anything, so any insights are welcome!
Input from Cory made me decide to go back and check that every DropDownList had only 1 selection specified and that's when I found the problem.
In getReservation() above:
        ' GET DRIVER
        ' -------------------------------------------------------------
        If dr("driverid") = "0" Then
            ' Self Drive
            rblTrips.Items(1).Selected = True
            pnlDriven.Visible = False
            pnlSelfDriven.Visible = True
        Else
            ' Driven
            rblTrips.Items(0).Selected = True
            pnlSelfDriven.Visible = False
            pnlDriven.Visible = True
            ddlDriver.ClearSelection()
            ddlDriver.Items.FindByValue(dr("driverid")).Selected = True
        End If
And then later:
        ' GET VEHICLES
        ' -------------------------------------------------------------
        ddlVehicle.ClearSelection()
        ddlDriver.Items.FindByValue(dr("vehicleid")).Selected = True
This was just bad copy/paste, so for people viewing this answer, make 100% sure that you're only selecting ONCE
As a side note, I'm told that the following is better practice:
' Do this:
ddlVehicles.SelectedValue = dr("vehicleid")
' Instead of this:
ddlVehicles.Items.FindByValue(dr("vehicleid").Selected = True
While any ListControl is being rendered, a flag is set if it encounters more than one selected item. If the flag is true, the abstract method VerifyMultiSelect is called. DropDownList's implementation of VerifyMultiSelect throws an exception if it encounters multiple selected items, so the only possible reason you're seeing this exception is because you have multiple ListItems with Selected set to true within one of your DropDownLists. 
I would carefully examine the results of your SQL to determine if there's a case where more than one item in the same DropDownList is having its Selected property set to true. There are three lines of code in the getReservation() function in your sample above that alter this property.
Perhaps it's the way you handle IsPostBack. There's a previously selected item stored in ViewState for that control, and you're trying to select another value along with it after post-back.
EDIT: OP found a solution.
I had a similar issue when reworking someone elses old code. For me it was a view state issue that was addressed simply by calling
myDropDown.ClearSelection(); 
They were pulling the datasource for the dropdown from an enum without a key and thus using FindText to make a selection. If the dropdown selected item was changed however they re-ran the statement on PostBack with the FindText without first calling
myDropDown.ClearSelection();
which as per Corys reply meant two items - the one just selected plus the one already selected in viewstate - were now selected and an exception was therefore thrown.
Not sure how it ever worked but it wasn't exposed until some minor change was made to code. So if anyone is struggling to find duplicate lines of code as per OPs answer, then you might also want to consider view state mechanics on post back.
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