I am sure there is something more to this phrase "One you have a
reference to the shape (example, in a varable named oSh)"

I managed to get this far but I don't know what I am doing (That's
being honest!).  There is still has a problem around the  "Select Case
oSh.Type".  Hence my suspicions above.

Does the Dim statement automatically set the Type0XCount to zero?

Sub Count_Object_Types_on_This_Slide()
   'Refers to each object on the current page and returns the number
of each Shapes.Type
Dim Type01Count As Long
Dim Type02Count As Long
Dim Type03Count As Long
Dim Type04Count As Long
Dim Type05Count As Long
Dim Type06Count As Long
Dim Type07Count As Long
Dim Type08Count As Long
Dim Type09Count As Long
Dim Type10Count As Long
Dim Type11Count As Long
Dim Type12Count As Long
Dim Type13Count As Long
Dim Type14Count As Long
Dim Type15Count As Long
Dim Type16Count As Long
Dim Type17Count As Long
Dim Type18Count As Long
Dim Type19Count As Long
Dim Type20Count As Long
Dim TypeUnknown As Long

For i = 1 To ActiveWindow.Selection.SlideRange.Shapes.Count
'Once you have a reference to the shape (example, in a variable named
oSh)
With ActiveWindow.Selection.SlideRange.Shapes(i)
ActiveWindow.Selection.SlideRange.Shapes(i).Select
       Select Case oSh.Type
           Case Is = 1
           'an AutoShape
               Type01Count = Type01Count + 1
           Case Is = 2
           'a Callout
               Type02Count = Type02Count + 1
           Case Is = 3
           'a Chart
               Type03Count = Type03Count + 1
           Case Is = 4
           'a Comment
               Type04Count = Type04Count + 1
           Case Is = 5
           'a Freeform
               Type05Count = Type05Count + 1
           Case Is = 6
           'a Group
               Type06Count = Type06Count + 1
           Case Is = 7
           'an Embedded OLE Object
               Type07Count = Type07Count + 1
           Case Is = 8
           'a Form Control
               Type08Count = Type08Count + 1
           Case Is = 9
           'a Line
               Type09Count = Type09Count + 1
           Case Is = 10
           'a Linked OLE Object
               Type10Count = Type10Count + 1
           Case Is = 11
           'a Linked Picture
               Type11Count = Type11Count + 1
           Case Is = 12
           'an OLE Control Object
               Type12Count = Type12Count + 1
           Case Is = 13
           'a embedded picture
               Type13Count = Type13Count + 1
           Case Is = 14
           'a text placeholder (title or regular text--not a standard
textbox)object
               Type14Count = Type14Count + 1
           Case Is = 15
           'a WordArt (Text Effect)
               Type15Count = Type15Count + 1
           Case Is = 16
           'a Media object .. sound, etc.
               Type16Count = Type16Count + 1
           Case Is = 17
           'a Text Box
               Type17Count = Type17Count + 1
           Case Is = 18
           'a ScriptAnchor
               Type18Count = Type18Count + 1
           Case Is = 19
           'a Table
               Type19Count = Type19Count + 1
           Case Is = 20
           'a Mixed object (whatever that might be)
               Type20Count = Type20Count + 1
           Case Else
           'a mystery!!! ?An undocumented object type?
               TypeUnknownCount = TypeUnknownCount + 1
       End Select
           MsgBox ("some message here")
   End With
   Next i
End Sub