Test or check if sheet exists

As checking for members of a collection is a general problem, here is an abstracted version of @Tim's answer:

Function Contains(objCollection As Object, strName as String) As Boolean
    Dim o as Object
    On Error Resume Next
    set o = objCollection(strName)
    Contains = (Err.Number = 0)
    Err.Clear
 End Function

This function can be used with any collection like object (Shapes, Range, Names, Workbooks, etc.).

To check for the existence of a sheet, use If Contains(Sheets, "SheetName") ...

You don't need error handling in order to accomplish this. All you have to do is iterate over all of the Worksheets and check if the specified name exists:

Dim exists As Boolean

For i = 1 To Worksheets.Count
    If Worksheets(i).Name = "MySheet" Then
        exists = True
    End If
Next i

If Not exists Then
    Worksheets.Add.Name = "MySheet"
End If

Some folk dislike this approach because of an "inappropriate" use of error handling, but I think it's considered acceptable in VBA... An alternative approach is to loop though all the sheets until you find a match.

Function WorksheetExists(shtName As String, Optional wb As Workbook) As Boolean
    Dim sht As Worksheet

    If wb Is Nothing Then Set wb = ThisWorkbook
    On Error Resume Next
    Set sht = wb.Sheets(shtName)
    On Error GoTo 0
    WorksheetExists = Not sht Is Nothing
End Function

If you are specifically interested in worksheets only, you can use a simple Evaluate call:

Function WorksheetExists(sName As String) As Boolean
    WorksheetExists = Evaluate("ISREF('" & sName & "'!A1)")
End Function