You know the phrase, “dress for the job you want”? Kind of like that. Spies need to dress appropriately for whatever their current cover is.
If that’s an official cover (someone who enjoys diplomatic immunity) then it’s probably going to be normal business attire.
For a non-official cover (someone who is not stationed in an embassy) it’s going to depend on exactly what their day job is. For someone who’s working as a lobbyist or corporate head hunter, then you’re still looking at business attire.
Also, forget about skin tight cat suits. Those are just about the worst thing a spy could wear. Nothing screams, “I don’t belong here” like an outfit that makes you look like a D-grade superhero reject. (Obviously, if you’re actually aiming for the superspy genre, then your character is a D-grade superhero, so you should plan accordingly.)
There are few things as embarrassing for a government as getting caught spying. Wearing an outfit that advertises covert action is a fantastic way to destroy your government’s soft power.
Usually, the justification given is that the spy/infiltrator/whatever is engaging in behavior that’s so dangerous, it won’t matter if they’re caught. But, that’s just an excuse to give a character cool toys. If your character is caught breaking into a government office in a turtleneck and jeans, the assumption will be that they’re a common criminal. If they’re breaking in with high tech IR goggles that can scan through walls, a three thousand dollar assault rifle, and a black cyberninja jumpsuit that blocks their own thermal signature, building security will know they just caught someone with serious backing. And the police have a lot more incentive to start peeling your character’s life apart until they find who sponsored them.
The other side of this is, a spy who has to revert to violence is doomed. (Not in the Sun Tzu sense; but they are not long for this earth.) Violence attracts attention. Attention makes it impossible for a spy to do their job effectively. Their job is social engineering, not playing James Bond.
Incidentally, even non-violent attention can also make your spy’s life a lot harder. A character who dresses to be the most attractive person in the room will find it much harder to slip away unnoticed. Especially if they’re trying to get away from someone determined to get in their pants. (Which we can add onto the pile of ways that James Bond as a wish fulfillment character sabotages his ability to function as a spy.)
Best case; your spy’s job is to get other people to break the law for them. Worst case; it’s to break the law in ways that will look innocuous until the last possible moment, and get out without anyone realizing something is off.
If you haven’t checked it yet, we’ve already written a fair amount on spies; some of that might be useful to you. Granted, our spy fiction recommendations do start to look fairly consistent over time. Also, given that the question started with (I assume) spy catsuits, you might also want to look at our stealth tag, it’s a lot shorter, but it might give you more useful information.
Burn Notice is still a really good primer on basic tradecraft. Pay more attention to the narration, Jeffery Donovan is, effectively, playing two different characters, and the Narrator is the one dispensing useful information.
The Sandbaggers is a little dated, and unfortunately expensive, but worth watching, for a more realistic look at spies. The show is based pretty heavily on the CIA’s special operation structure, rather than MI6′s. But, otherwise it is still worth watching.
Queen & Country by Greg Rucka is a modern homage to The Sandbaggers. The comic loses a lot once you’ve actually viewed the source material Rucka was pulling from. But, it is also a lot cheaper and easier to find.
In spite of (basically) being a James Bond fanfic, The Bourne Identity by Robert Ludlum actually has a surprising amount of useful observations buried in there. Ludlum takes pains to explain the ways Bourne blends into his environment. Some of this is fairly obvious, but it’s worth seeing in action anyway. The films are entertaining, but not particularly useful, however. Also, I could never really get into Supremacy, so I don’t know if the later books are worth looking at.
John Le Carré’s Tinker, Tailor, Soldier, Spy is the opposite of Bourne (and Bond, for that matter). If you’re wanting to write a spy that actually manipulates the people around them, Le Carré’s work is something that needs to be on your radar.