Finding a string should have a particular letters?

The same you can achieve using Linq[^] query:

string[] words = new string[]{"Anitha", "Nandu","anushka","jayasree"};

var qry = from w in words
        where w.ToLower().Contains('a') && w.ToLower().Contains('n') && w.ToLower().Contains('u')
        select w;

foreach(string word in qry)
{
    Console.WriteLine(word);
}

Result:

Nandu
anushka

How about this - not RegEx, but produces also the desired result:

var names = new string[]
{ "Anitha"
, "Nandu"
, "anushka"
, "jayasree"
};
string[] anu = names.Where(n=>n.IndexOfAny("aA")>=0)
                    .Where(n=>n.IndexOfAny("nN")>=0)
                    .Where(n=>n.IndexOfAny("uU")>=0)
                    .ToArray()
                    ;

Cheers
Andi

See here..

The 30 Minute Regex Tutorial[^]

Your requirement is not totally clear.
A. Do you just want to find any string that contains at least 1 each of a, n, u?
B. Do they have to be in the order of something, then 'a', then later an 'n', then later a 'u'?
Of your 4 examples, it appears that 1 and 4 should fail, and 2 and 3 should pass.
Right?
For the case A above:

(?:a.*n.*u)|(?:a.*u.*n)|(?:n.*a.*u)|(?:n.*u.*a)|(?:u.*n.*a)|(?:u.*a.*n)

For case B:

(?:a.*n.*u)

The / delimiters and i at the end for case insensitive depend on the context where you are using this. For example, these would be appropriate for a Perl script, but not for a .NET Regex.

I would suggest two possible solutions:

1) Using regex: (using positive lookahead)

Dim names As New List(Of String)(New String() {"Anushka", "Anitha", "Nandu", "jayasree"})

'Solution 1
For Each name As String In names
    Debug.Print(System.Text.RegularExpressions.Regex.Match(name, "^(?=.*a)(?=.*n)(?=.*u).*$").ToString())
Next

2) Using loops and Lists:

Dim names As New List(Of String)(New String() {"Anushka", "Anitha", "Nandu", "jayasree"})

'Solution 2
Dim mustHaveChars As New List(Of Char)(New Char() {"a"c, "n"c, "u"c})
For Each name As String In names
   Dim n As New List(Of Char)(name.ToCharArray().Distinct().ToArray())
   If n.Except(mustHaveChars).Count = n.Count - mustHaveChars.Count Then
      Debug.Print(name)
   End If
Next