In my personal opinion, signatures are a core element of online forums, as it allows people to show their personality (Stated above this post), and gives more color to the threads (As opposed to having only words in the pages).
Restrictions to what signatures can have? For sure. Can't have too many images on one signature, GIFs, or autoplay videos (As they mess with the bandwidth, also stated above), or even sounds from that dreadful 404 Time Not Found website. (Older members definitely know what I'm talking about here xD)
I'll vote to maintain signatures on the forums