If you have a wet/dry vacuum, you can also just suck out the water and the clog. This works well for shower or tub drains that are likely clogged more by hair than oil-based substances.
Got a plastic P trap under that sink drain? You can unscrew the fittings by hand, no wrench needed. Catch the water in a small bucket and clean it out by hand. Screw everything back on by hand. Take pictures when removing the fittings if you're nervous about putting it back together
No idea why I didn't think of just removing the whole plastic piece. I always wondered why they never had a cleaning point, but why have one when they come off, I guess.
Handy too if your water is turned off and you have a few gallons set aside then you can use this technique to flush. Works best out of a pitcher or something with a wide mouth so all the water comes out in one shot. If just using to flush, the water doesn't have to be hot.