Xenforo is the forum software most incel forums use as of 2019. It is extremely good looking for a forum software, perhaps justifying its license price of $160 + $40 yearly renewal. Incels.co notably uses it.
The forum software is modeled after legacy forum software such as VBulletin making it familiar to traditional users of forum software. It has fairly high server overhead compared to myBB type forums, but remains functional. It is written in PHP.
It does not try to change the paradigm of forum software, instead trying to build off the past and make it look good. Lately it has been borrowing from Facebook in an attempt to modernize.