Add class to Menu Block UL unique to each Menu Block instance? I've seen lots of answers, but they only allow you to add classes to all Menu Blocks, or to specific Menus.

I have multiple menu blocks with the same menu behind them (1st level block, 2nd level block, etc.)

I need each menu block to have a unique class based upon the name of the Menu Block. Is this possible?
Unless your theme changes the usual behavior then each of your menu blocks will have its own unique ID so you can use CSS like this (note the ID/class I used here might not be the ones you have to use, they're just an example):

#block-menu-block-2 ul.menu


This way you don't have to worry about overriding the markup of the menu to add classes in.