Mike,

RichEditBox supports methods:
GetCharIndexFromPosition and GetLineFromCharIndex.

Combining both you can get what you want:
GetLineFromCharIndex(GetCharIndexFromPosition(pos))

Be advised though that the line index is the physical line on the scree. If
one logical line is wrapped over more than one physical line this method
will return different line indexes depending where you click on the line.
From my experience this is almost never what one expects. Unfortunately the
only way to get the logical line index (without writing any code) is to turn
off the text wrapping.
This is not only missing functionality in the .NET control it cannot be done
with win API either.

I think I got it.

   Private Sub rtbResults_MouseDown(ByVal sender As System.Object, _
                   ByVal e As System.Windows.Forms.MouseEventArgs) _
                   Handles rtbResults.MouseDown
       If e.Clicks = 1 And e.Button = Windows.Forms.MouseButtons.Left
Then
           Dim positionToSearch As Integer = _
               rtbResults.GetCharIndexFromPosition(New Point(e.X,
e.Y))
           Dim lineIndex As Integer = _
               rtbResults.GetLineFromCharIndex(positionToSearch)
           Dim lineString As String = rtbResults.Lines(lineIndex)
           ' do whatever
       End If
   End Sub

My goal here was to highlight filenames and display them as links and
react upon them when clicked.  Although the RichTextBox will create a
link for files, it stops at the first space it encounters, rendering it
absolutely usless for the task.

This is what I use to highlight the files in the rtf box.  If any
flaws, please update and post a reply.

    Private Sub HighlightDirs(ByVal rtb As RichTextBox)
       Dim r As Regex = New Regex( _
       "^([a-zA-Z]\:|\\\\[^\/\\:*?""<>|]+\\[^\/\\:*?""<>|]+)" & _
           "(\\[^\/\\:*?""<>|]+)+(\.[^\/\\:*?""<>|]+)$" _
           , RegexOptions.IgnoreCase Or RegexOptions.Multiline _
               Or RegexOptions.Compiled)
       For Each m As Match In r.Matches(rtb.Text)
           rtb.Select(m.Index, m.Length)
           rtb.SelectionFont = New Font(Me.rtbResults.Font, _
               FontStyle.Bold Or FontStyle.Underline)
           rtb.SelectionColor = Color.Blue
       Next
   End Sub