Looking at images of the Olympus Trip 35, the hotshoe is a standard ISO one, with the single sync contact, so any ISO-compatible hotshoe flash (i.e., pretty much all of them) should fire correctly in sync. But they won't do anything else. You'll have to use the flash in Manual mode, and manually dial in the power level you want, or hope you find a flash that includes autothyristor technology, if you want automated power. TTL, HSS/FP, remote commanding, etc. are all features you can kiss goodbye with a manual-only hotshoe.

The main problem you're going to have is finding a tiny flash that has full manual mode on it. Most of the very small very cheap flashes that are basically as effective as a pop-up flash are typically TTL-only, and don't offer a physical way to control the power level of the flash's output on the flash itself, but rather relies on TTL communication through the hotshoe's non-sync contacts--which the Trip 35 doesn't have.

The only real candidate I can think of would be a Nissin i40. But then, my knowledge of small-size hotshoe flashes isn't great.

Any new digital-compatible flash will have a very low sync voltage (typically less than 5V, but definitely under 10V), so I wouldn't worry about frying your camera's hotshoe--it's usually only when you put a film era speedlight on a digital camera that you run that risk.