Xenforo is the forum software most incel forums use as of 2020.
It's extremely good looking for forum software, perhaps justifying it's initial license price of $160 + $40 yearly renewal.
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.
Most active incel and pseudo-incelospherian forums use Xenforo as of 2020. Also none of them use the stock theme, opting for paid, free, or custom alternative themes. Which is a bit odd, given the default theme is quite good.