Other guys will jump in.
It depends on the cufflinks - and the shirt.
There are SO many jillions of cufflink designs. Very simple black links are usually for more formal wear or evening. The fun ones are the novelties. Little cigars, automobiles, hats, exotic themes, horses, dogs, political, club affiliation, you name it. Those are for other times and day wear.
Most shirt studs one sees are onyx or mother-of-pearl. Those go with starched shirts in a formal setting. If you find any "novelty" shirt studs... well... I'd think a time or six about those..
Am I wrong, gentlemen?