Replace non-numeric with empty string

Definitely regex:

string CleanPhone(string phone)
{
    Regex digitsOnly = new Regex(@"[^\d]");   
    return digitsOnly.Replace(phone, "");
}

or within a class to avoid re-creating the regex all the time:

private static Regex digitsOnly = new Regex(@"[^\d]");   

public static string CleanPhone(string phone)
{
    return digitsOnly.Replace(phone, "");
}

Depending on your real-world inputs, you may want some additional logic there to do things like strip out leading 1's (for long distance) or anything trailing an x or X (for extensions).

Here's the extension method way of doing it.

public static class Extensions
{
    public static string ToDigitsOnly(this string input)
    {
        Regex digitsOnly = new Regex(@"[^\d]");
        return digitsOnly.Replace(input, "");
    }
}

You can do it easily with regex:

string subject = "(913)-444-5555";
string result = Regex.Replace(subject, "[^0-9]", ""); // result = "9134445555"

You don't need to use Regex.

phone = new String(phone.Where(c => char.IsDigit(c)).ToArray())