I've seen a ton of players using hair scrunchies, or similar soft bands to mute, or rather dampen strings.
What are some of the best ways to do this? What works better vs what doesn't mute very well?
All depends on personal style. You can use the ties to mute strings or overtones you don't wish to ring, in which case the applicable tension depends on how hard you play. You can also use a product like the GruvGear Fretwraps to provide adjustable tension. I find muting overtones also works well when tracking bass-to-MIDI. Victor Wooten also uses a technique where he moves the tie to a certain spot on the neck and plays below it, producing an artificial harmonic.
It's all experimentation, though, really. Have fun!