Film Advance Lever and Shutter Curtain

Hit or press the film release button or replace with a known good battery.

Is there film in the camera, if so rewind the film and then test the empty camera, if your problem persists, service the camera.

A Canon AE-1 is not an EOS camera the Canon "A" series is manual focus while EOS is auto focus. First thought In answer to the advance lever stuck or jammed I'm going to assume the camera is in serviceable condition and has not been subjected to water damage or harsh use. If we are dealing with just the advance lever then I'd say that you have over advanced the film at the last frame. I've done this myself after loading a 24 exposure and thinking I had a 36 exposure film. At the bottom of the camera there is a little pin which releases the advance lever and allows the user to rewind the film. Under normal conditions when it's time to rewind the film into the canister this pin offers little resistance to pushing in to release, however depending on how much force the advance lever has been subjected to increases the tension on this pin. Push the pin in and rewind the film.
Second thought is if you have film in the camera and it has not been used fully then it's possible that the film has already been advanced to a new frame and the shutter needs to be activated before the lever will become free and advance to the next frame.
Other problems with the Canon "A" series cameras is they have a tendency to dry out the lubrication and begin to squeak when the shutter is fired. This condition can eventually result is a shutter seizure and give the same advance lever condition.
Those are the three things that come to mind over this lever problem lets hope it's just run out of film.

You can check the shutter without film in the camera. Remove the lens and open the film door. You will see the cloth curtains of the shutter in the center. Advance the film lever to **** the shutter and set the speed dial to a slower speed like 1/15. Hold the camera so it is aimed a a light source, (a window will do) and fire the shutter. You should see the light through the shutter for the time duration is is set to. (1/15). If you see the light source, repeat the process going up one speed setting until you reach 1/1000. If you see the shutter open and close as described, then try another roll. After you load the film and close the back door, make note of what the rewind knob is doing as you advance the film. It should spin counter-clockwise each time. That is a easy test to know that the film is traveling in the camera properly. Vary your exposures with the new roll at different speeds and f-stops. You should get exposures, but if not let us know.

This is not the best solution as it indicates a faulty film back. The film should advance without resorting to activating the film-wind release lever. That is only for intentionally advancing a partially exposed roll of film. Make sure the multi-exposure lever is not engaged and that the counter does advance and the red mark disappears as you advance to an unexposed frame.

needs repairing at a qualified shop
when was it last cleaned or serviced ?
if you think the camera is worthy, since its old how much is it worth to you vs a new one

It likely did not catch when it was first loaded, and as such never advanced through the camera. Open the back of the camera, with no film in it, then set the shutter speed to 1, advance the film advance lever and take a shot. You should see the shutter curtains open and then close a second later. Then advance the film lever again and watch to see that the sprockets are turning. If they are, then the camera is exposing and the mechanics are working properly to advance the film. Chances are it was just loaded incorrectly.

In most film cameras, a battery is not necessary to manual advance film using the lever. Your Canon AV-1 is no exception. I have owned two cameras from the same group, the AE-1 and the Canon A-1. Neither required a battery for manual film advance.

Make sure the ring around the shutter release button is set to "A", otherwise the shutter won't release and the film lever won't work correctly. If the ring shows an "L", the shutter release is locked.

Here's a graphic of the batteries that can be used in your AV-1:

Yes this is fairly common with K 1000 and the Spotmatics : although the shutter is cocked and hence you cannot advance film you cannot release shutter because mirror box is not cocked. Remedy : remove bottom cover and locate mirror box lever which protrudes slightely and is movable up and down and is spring loaded. Move upwards until it licks in position ( moving up against spring tension ). Now release the shutter and camera should let you advance the film .