Which perks stack and of those which stack, how do they do so?

This is not an exhaustive list, and it's subject to change. It's just what I've found so far.

Greed stacks multiplicatively with itself and Trick Greed. Two copies of Greed make each nugget worth 40 instead of the 30 you would expect from additive stacking.

Trick Greed seems to make more nuggets, but I can't tell how it stacks. I had Attract Gold at the time, and it made counting difficult.

Strong Levitation stacks, but without numbers to check it's hard to tell how. One downside is that it doesn't increase how fast your levitation meter refills. So the more copies you have, the longer you have to wait for it to fill.

Faster Movement definitely stacks as of February 16, 2020. Not sure if it's multiplicative or additive. Not sure if Faster Swimming stacks.

Perk Lottery stacks multiplicatively. Each copy improves your odds of the other perks remaining, but there's always a chance of losing them. It's never a guarantee.

Extra Perk always adds one perk to the list. Be careful, picking it too many times makes it hard to pick the perk you want. The pedestal gets crowded and it's easy to accidentally grab the wrong one. Picking it twice is about the limit if you want to keep it easy.

Personal shield seems to stack. I can't guarantee it, but it does feel more effective. Maybe it's just wishful thinking.

As of February 16, 2020, Don't Skip Leg Day definitely stacks. I have 3 copies of it, and when I kick exploding boxes they fly away crazy fast. They're off the screen in just a few frames, even if they bounce off a wall along the way.

Extra Health per Heart doesn't stack, though I'm told it used to. Maybe it will again some day.

As of October 21, 2020, More Love stacks. 4 copies makes medic hiisi heal you.