I love Skuli and Stomu! I sometimes share the bass spot with Stomu in the Nublu Orchestra. He usually uses lots of crazy effects while I just plug right into my amp. They're both great guys!
I would say that some of the techniques that could be considered 'avant' would include:
Moving the strings with something besides a plastic/metal plectrum or your fingers. This could include the E-bow, 'funk fingers' (wooden dowel finger extensions), wooden hammers ala hammered dulcimer and 'prepared' bass where you put various objects in the strings' path and use various materials to put the string(s) into motion. Examples of prepared bass are Skuli Sverrisson of course, "Paper Bass" by MMW and "Buzz" (from Medicine Wheel) by Ben Allison.
Playing a string while it's being pulled (or pushed) off the edge of the fingerboard. Chris Wood and Victor Wooten, among others, do this.
Playing with a slide. Very common on guitar but not so much on bass. RIP Mark Sandman!
Pressing the string down between the end of the fingerboard and the neck pickup. Regi and Victor Wooten do this a lot.
Playing the non-vibrational (non-critical?) parts of the string (behind the nut or behind the saddle/bridge). Although that's the term that's commonly used, those parts of the string do, in fact, vibrate.
Sympathetic vibration. By setting the bass' wood in motion you can get the strings moving sympathetically.
Tapping the bass' body like a drum. Much more effective on DB but possible with EBG, too.
The 'free-stroke' method ala Willis, Garrison. This is very similar to classical guitar technique and can involve anywhere from one to all five fingers of the right hand. This is probably the most visible and exciting new technique, although it is anything but new.
Using thumb position on electric bass (fretting hand). Notable users of this technique include Steve Bailey. I've seen Patitucci do it, too.
Double-thumb or 'double-thump' technique. Popularized by Victor Wooten.
Thumb-slapping on DB. Not to be confused with regular ol' slappin'. Stanley Clarke does this thing on Rite of Strings where he plays with his thumb right in front of the bridge and he gets a really cool sound. I would equate it to electric thumb slapping technique on DB although his thumb is facing into the center of the bass rather than parallel with the E string or facing away from the bass like it would on EBG.
Using an apparatus built into the bass to strike the strings ala Neuser Claudia
http://www.neuserbasses.com/claudia.html Playing from underneath ala Spector video dude. I've seen a couple people play this way before. Crazy stuff!
Yes, Mike Watt's "flipper" technique definitely qualifies as non-standard! He does all kinds of stuff. When I lived in Lawrence, KS for ten years, I saw him every time he played at the Bottleneck. I even did sound for him there a couple of times. Super cool dude!
These are just off the top of my head. I suppose there are as many as our imaginations will allow!