Mutes _ When and why have you used one on your bass ?

I've used a sponge slice as a mute before, but don't need one much these days. But when I do, it's usually a palm mute and a pick.

    I use them occasionally, it kind of helps the open notes conform a bit with the fretted ones, particularly when the treble is being tweaked and the bass line is moving. Helps a bit in uniforming sustain as well. Particularly when a lot of flags are hanging off the post ♪

    Hardly matters when everybody’s cranked at 12 in a 10,000 square foot dive “o) 

Just enough to stay in place works for me.

Back in the ‘80s I picked up a video of TOP’s Rocko Prestia to learn his style of fretting and muting with the same hand and found it a real challenge. I don’t know if it was ever offered as a dvd or download so it may not be possible to find these days.

    Yes, and it’s not exactly square but close enough to hold it in place, I can twist it 90° and it slides out. I can use palm muting too, but when I’ve got all fingers working the strings I can’t palm them effectively. My right arm is a mess from various accidents over the years. The wrist will not twist, the radius bone of my forearm is fused to my elbow, along with the bicep muscle being paralyzed. The fingers all work fine, though it took a few years getting them to work properly. When I broke my neck I severed a group of nerves that worked the arm, at first I couldn’t do much with my fingers or lift my arm at all. An earlier accident shattered my right arm elbow, that’s when the radius was fused. 

Here's a clip of Bob Babbit playing with a mute. At the end of the clip you can see what looks to be a foam mute by his bridge.

I hadn’t thought about that, I guess I must of at one time though, I usually mute the strings I’m not actively playing with my fretting hand, likely because of exactly that. Even when everyone is going direct with headphones playing the bass itself will vibrate strings not actively being used. Maybe one of the reasons I’ve never taken to the five string. Have one and used it, but still I prefer the 4 string. 

The five string was making me lazy, I could cover just about everything at the 5th fret position. I was starting to Play less of the neck. My fault not the instruments.

I have a couple of older basses with built in mutes , I rarely dial them in . Many friends use sponge mutes for that old school thump .
It’s funny how everybody goes on about sustain one minute , and muting the next .
I’m fussy about note length , and don’t care much for sustain .