score:0

Accepted answer

An approach

    Dim list As New List(Of String) From {"a", "e", "e", "b", "c", "c", "b", "c", "d", "e"}
    Dim duplicates As IEnumerable(Of String) = list.Distinct
    duplicates = From s In duplicates
                  Where list.FindAll(Function(l) l = s).Count > 1
                  Select s

    For Each s As String In duplicates
        Dim ct As Integer = (From l In list Where l = s Select l).Count
        Debug.WriteLine(s)
        Debug.WriteLine("count:{0}", ct)
        Dim idx As IEnumerable(Of Integer)
        idx = From n In Enumerable.Range(0, list.Count)
              Where list(n) = s
              Select n Take ct

        Debug.WriteLine("Indexes")
        For Each n As Integer In idx
            Debug.Write(n.ToString)
            Debug.Write(" ")
        Next
        Debug.WriteLine("")
    Next

Output

e
count:3
Indexes
1 2 9 
b
count:2
Indexes
3 6 
c
count:3
Indexes
4 5 7

score:2

You already have an answer, but I propose a slightly different method anyway, it might come in handy after. This uses the same LINQ methods you already have.

The LINQ's query creates an object that contains the value and position of all strings in the elements List, groups the values then selects the groups that contain more than one item and finally orders the groups by their Key value. 

Dim elements = {"a", "b", "c", "c", "b", "c", "d", "e", "e"}.ToList()
Dim duplicates = elements.Select(Function(elm, i) New With {.Value = elm, .Index = i}).
                          GroupBy(Function(obj) obj.Value).
                          Where(Function(grp) grp.Count() > 1).
                          OrderBy(Function(grp) grp.Key)

For Each group In duplicates
    For Each dupe In group
        Console.WriteLine($"Duplicate: {dupe.Value} Index = {dupe.Index}")
    Next
    Console.WriteLine($"Number of {group.Key} duplicates: {group.Count()}")
Next

This prints: 

Duplicate: b Index = 1
Duplicate: b Index = 4
Number of b duplicates: 2

Duplicate: c Index = 2
Duplicate: c Index = 3
Duplicate: c Index = 5
Number of c duplicates: 3

Duplicate: e Index = 7
Duplicate: e Index = 8
Number of e duplicates: 2

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