FilterStringContainsNew.cs

// Contains
public static bool ContainsIgnoreCase(this string value, string substring)
{
    if (string.IsNullOrEmpty(substring))
        return false;

    return value.IndexOf(substring, StringComparison.OrdinalIgnoreCase) >= 0;
}

// Equals
public static bool EqualsIgnoreCase(this string value, string match)
{
    if (string.IsNullOrEmpty(match))
        return false;

    return string.Equals(value, match, StringComparison.OrdinalIgnoreCase);
}

// Anwendung
var myList = new List<string>();

var result = myList.Where(w => w.ContainsIgnoreCase(searchstring)).ToList();