Regex for checking if a string has mismatched parentheses?

Regex is not the right tool for the job. Scan a string manually.

Pseudo-code:

depth = 0
for character in some_string:
    depth += character == '('
    depth -= character == ')'
    if depth < 0:
       break

if depth != 0:
   print "unmatched parentheses"

You could use:

^(mailto|ftp|joe)

But to be honest, StartsWith is perfectly fine to here. You could rewrite it as follows:

string[] prefixes = { "http", "mailto", "joe" };
string s = "joe:bloggs";
bool result = prefixes.Any(prefix => s.StartsWith(prefix));

You could also look at the System.Uri class if you are parsing URIs.

The most efficient regex approach is to use the principle of contrast, i.e. using opposite character classes side by side. Here is a regex that can be used to check if a string has 3 Latin script letters or digits:

^(?:[^a-zA-Z0-9]*[a-zA-Z0-9]){3}

See demo.

In case you need a full string match, you will need to append .* (or .*$ if you want to guarantee you will match all up to the end of string/line), but in my tests on regexhero, .* yields better performance):

^(?:[^a-zA-Z0-9]*[a-zA-Z0-9]){3}.*

Also, a lot depends on the engine. PCRE has auto-optimizations in place that consists in auto-possessification (i.e. it turns the * to *+ in (?:[^a-zA-Z0-9]*+).

See more details on password validation optimizations here.