If you grew up with the NES, you remember that their basslines had a certain "snappiness" to them (for lack of a better word). Here are two of my favorite examples:

DAT BASS!

Anyway, I've been trying to mimic that type of bass as closely as I can in my PICO-8 carts, but I haven't quite hit the mark yet.

Does anyone have instrument/effect suggestions for achieving this kind of sound?

The NES's sound chip, the 2a03, has five channels. Two square waves, a triangle wave, a noise channel, and a sample channel. Both of these examples, and most NES games in general, use that triangle channel for bass, and wouldn't you know, Pico-8 also has a triangle wave instrument, which is instrument 0.

[sfx]

Here I've recreated a bit of that Urban Champion song. It's not exact of course, but doing mostly short, low notes with the triangle wave like this is just about the closest you can get to it. the 2a03 has it's own list of weird little quirks just like the Pico-8's sound chip does, so you can't really get it 100%.

[sfx]

You could also shift it an octave lower, which would sound like this, but to me that's a bit less "snappy" sounding.

If you'd like to play around with the NES's actual sound chip for yourself, I highly recommend checking out Famitracker. If you're already used to Pico-8's tracker it shouldn't be too hard to figure out. And if you want to approximate the 2a03 in Pico-8, the best way to do it would be to only use instruments 3 and 4 on channels 2 and 3 (interchangeably, as in either track can use either instrument at any time), use instrument 0 on channel 3, and instrument 6 on channel 4. I know a decent bit about the 2a03 and about music in general so feel free to ask me any more questions you might have.

These are beautiful! Thank you!

The difference in the bass sounds here is in that overtone you hear on the pure triangle wave sound of the NES. When the wave hits the peaks it quickly snaps your speaker in the other direction, causing an overtone to be apparent in your ear. Pico-8's triangle wave seems to be a bit more smoothed out, and so sounds closer to a pure sine wave.

You could have another instrument mirroring the notes, but a few octaves up and much quieter to get a similar overtone sound.

If I can figure out how people are posting these players, I might try to do this method myself and post here...

The bbs creates these players automatically when you paste SFX or music copied from pico-8! (they are copied in a text format that the bbs recognizes)