凭 + [noun] + [verb phrase]
[subject] + 凭(着)+ [noun] + [verb phrase]
[subject] + 凭什么 + [verb phrase]?
凭 + noun names the basis for an action: the document that lets you in, or the quality that gets you something. In notices it's terse and formal.
With a person's ability or effort it says what got them there. 凭着 is a slightly fuller form.
只凭 says the basis is too thin.
凭什么 challenges someone's right to do something. It's spoken and blunt.
用 is for a tool or a means you physically use. 凭 is for a credential or a quality you rely on.