alt Hacker NewsIt does happen, but it's way less lucrative. Tends to be limited to actual damages rather than punitive damages. There have been some scam-ish sub-industries (fake whiplash claims, suing councils for tripping over cracks in the pavement). It's very rare to see advertising for lawyers.
It's also rare because advertising for lawyers (and doctors) is strictly regulated in some member states. A sign in front of the office saying "S. Goodman, attorney, specialized in drugs, organized crime and whiplash" is OK, billboards, TV spots, newspaper ads and any kind of claims beyond "I'm an attorney and this is my office and specialty" are verboten.