Most superheroes past and present are outfitted with a variety of ridiculous outfits, some practical and others frivolous. Other crime fighters traveled a more practical path, choosing to forgo the ritual of changing clothes before engaging their enemies. Here is a list of some superheroes who wore normal clothes.
Luke Cage is on the left in the above image, apparently conducting an interview with Ghost Rider. He doesn't wear that yellow shirt get-up anymore. Some would argue that Cage's jeans and t-shirts are so tight they should qualify as a costume. In the comic world though, jeans and a t-shirt provide much more coverage that the typical outfit regardless of how tight they may be. One reason he probably doesn't wear a costume is that he was a pretty normal person before prison experiments gave him his powers.
Powers: Cage is super strong and agile. His real claim to toughness is his nearly impervious skin. Very few weapons can pierce his outer layer, including bullets and vampire fangs.
It's true, Xavier has been featured in various types of X-Men garb. Still, the vast majority of his appearances show him in typical 'professor-type' clothes: slacks, loafers, tie, suit jacket and/or a fashionable vest. Xavier is an incredibly powerful telepath so it is possible that he has been in costume all along, simply controlling our minds to believe he buys all his clothes at Brooks Brothers.
Powers: Xavier is one of the most powerful telepaths in his comic universe. He lost the use of his legs and is most commonly seen floating around in his customized wheelchair.
You could argue here that the Punisher (aka Frank Castle) does wear a costume: his signature black tee with white skull. Still, in the comic world these are street clothes regardless of the recurring logo. Castle is one of the darkest 'heroes' and one of my favorite given his penchant for simply killing every bad guy he sees. Like a gun-happy Dexter. Given his 'dark alley' reputation, Castle is typically found sporting dark jeans or pants with his typical shirt.
Powers: He doesn't have any super powers but he does have an uncanny ability to get his hands on pretty much any hand-held weapon known to man.
The Incredible Hulk
Green skin and ripped jean shorts is no costume, merely a cause of intense and instantaneous growth and a natural love of denim. Bruce Banner is normally caught off guard when the time comes for him to burst into a giant, muscular Jolly Green Giant. Luckily, a small portion of his jeans always remained to keep him compliant with the Comics Code.
Powers: Banner isn't known to control his powers well, but it's clear that the result of his mutation is to grow quickly into a giant, super-strong and resistant humanoid.
There are a few different variations of Ghost Rider in the comic universe, Daniel Ketch and Johnny Blaze being the most prominent. They both possess similar clothing features: jeans or dark colored slacks, t-shirt, leather jacket, riding boots, and sometimes chains. Again, this is a super hero who, while he wears the same outfit over and over, it is not a traditional hero get-up.
Powers: Ghost Rider possess' the super power of a flaming skull for a head, super strength (like, throw a garbage truck into a skating rink strong), a motorcycle that leaves a trail of flames and can ride on pretty much any surface including water and vertical walls. Also, Ghost Rider is a ghost of sorts.
This list is pretty short which gives you an idea of how frequently writers craft plain-clothed characters. If I left anyone off this list of super heroes who don't wear costumes, let me know in the comments section.