The problem here is, there is no “typical handgun.” At the extremes, the M1911′s safety locks the trigger, slide, and (if I’m remembering correctly) hammer. So you can’t operate any of those mechanisms while engaged. If the safety’s on, you can’t chamber a round, pull the trigger, or even cock it.
On the other end of the spectrum, you have weapons like the Glock series pistols, Walther P99, S&W Sigma, and a number of other pistols with no external safety whatsoever. You can’t safe a Glock. There are internal safeties to prevent accidental discharges, but there’s no switch that will disable the trigger.
Similarly, SIG P220 series pistols lack external safeties. These vary between single action variants, where the closest thing to a safety would be carrying the hammer down (slide blowback recocks the pistol between shots), and double action ones, where the trigger pull is heavier on a decocked hammer, but, again, there is no safety, just safe weapon handling practices.
In contrast the Beretta 92 actually has a safety lever. It can be either mounted on the slide or the frame, depending on the age of the pistol.
Like the Beretta 92, the H&K USP has alternate safety configurations. Ten of them. Externally these are the same pistol, with a single clearly identifiable safety lever (usually) mounted on the left side of the frame above the thumb. (About half the variants swap the lever to the right side of the pistol, along with the slide release for left handed shooters.) What the lever does varies based on the specific (externally identical) model. On some USPs it functions as a manual safety, on others it can decock the hammer, or both. Including a variant that will decock the pistol if you pull the trigger while the safety is engaged.
Depending on the pistol, the safety can be mounted on the frame, near the thumb, near the front of the frame, on the slide (usually behind the chamber), on the trigger, on the trigger guard. This is assuming the pistol has a manual safety at all, which isn’t a certainty.
Beyond this, grip safeties along the back of the magazine, and trigger mounted pressure safeties, are somewhat common. These disengage when the pistol is held in a firing position, and are specifically designed to prevent a shooter from finding they accidentally left their safety on. Which is actually the intent behind the Glock and P99′s design (among others).
Someone who’s unfamiliar with firearms in general, looking for a safety (assuming it has one, and it’s not clearly marked) is almost as likely to eject the magazine, or begin the disassembly procedure, depending on the pistol.
Really, with questions like this, your best bet would be to look up the kind of pistol you’re wanting to use, and reading up the operation methods for yourself, so you can then know exactly how your character will mishandle it. Wikipedia is a pretty good starting place to decide on a gun.
If possible, I’d also strongly recommend you spend some time on the range with the pistols you intend to give to your characters. Many ranges will rent guns to shooters, and you’ll probably end up spending around $100 for the experience (and ammo).
They may require you to take a firearms safety course, in which case, do it. It’s information you need. Also, take it seriously. We’re talking about you actually handling real firearms, these are not toys. Depending on range policies, you may need to bring along a friend.
It’s not always possible to fire the exact gun you have in mind. If you’re giving one of your characters a Glock 20 or Makarov PM, don’t automatically expect to find one at the local range. In general, firearms that use exotic or rare calibers are going to be harder to rent than ones that use common ones.
If you’re going to use a specific gun, you need to know how it works, even if your characters don’t.