So far I have found four airlines that say in their policy that they don't (intentionally) overbook flights. I will add to this list (and invite other to do so) if I find more.
As @chx said, Ryanair's Passenger Charter explicitly states they do not overbook flights. Of course you can be bumped for other reasons like if you cause a disturbance etc or if a flight crew need an extra seat (this is very rare).
From personal experience: this is true. I fly Ryanair a minimum of 4 times a year and I've never had seen or heard of a passenger being kicked off a plane. They don't offer stand by tickets and free check in is only 30 hours before the flight. The only allow online check so whether you pay to choose your seat or not, you have a seat number before you arrive in the airport.
This airline does not "intentionally" overbook flights but still has policy in place for "accidental" overbooking:
While JetBlue does not intentionally overbook its flights, there is still a slight chance that a seat will not be available on a flight for which a person has a confirmed reservation. If the flight is overbooked, no one will be denied a seat until airline personnel first ask for volunteers willing to give up their reservation in exchange for a payment of the airline’s choosing. If there are not enough volunteers, JetBlue will deny boarding to other persons in accordance with its particular boarding priority.
There have been reports of many over bookings by JetBlue in spite of this policy (hence the inverted commas). One claim
I have not been able to access SouthWest's site to confirm their policy but this news article claims that:
CEO Gary Kelly announced Thursday that Southwest (LUV) will no longer overbook its flights
WestJet is another airline that claims it does not overbook its flights saying:
We’re proud that we do not intentionally oversell our flights – we only sell the same number of seats that we have available on an aircraft. However, there are instances, operationally, where for example we need to switch to a smaller aircraft, or there may be a mechanical issue with a seat, that can create a situation where a flight becomes overbooked. In these cases, we’ll work with our guests, and ask for volunteers willing to take a later flight in exchange for compensation.
You can take it that all airlines overbook until proven otherwise. It's a legal and common practice. In general, I haven't seen many reports of people being forcibly removed from the plane in Europe because European Airlines supposedly "use algorithms that predict the number of no-shows per route". Check out this article if you're more interested.