With an airspeed of 0, I would suspect a porous blockage -- likely various bug bits. Mud dauber wasps block off the pitot completely. I once was on a takeoff run on a short runway that ended with a downslope to a ravine when I noticed that the ASI had not come alive; so, took the problem into the air.
I talked to ATC and got some groundspeeds, then decided to try some stalls at higher altitude. As I climbed the ASI went up -- and came down when I descended. Basically the ASI had become an altimeter
There was not much daylight left and ATC offered to take me in at the big airport, but I figured I had enough time to try a landing, flew the power settings I usually do and landed just fine.
Once you are used to flying a particular airplane, You can generally do fine without the ASI.