For the common case of an LCD screen, almost any film but use a tripod, and turn the screen brightness down very far if you want to see the surroundings. As comments have pointed out, film balanced for the color of incandescent light (tungsten) is going to have the wrong white point for a computer screen. These are unusual these days and will almost universally have a T somewhere in the film name.

You aren't going to find any film fast enough to let you take pictures indoors handheld in 35mm with decent quality, so don't try for that.

If you were using an SLR you might be able to use a flash that was off the camera to try to fill-in the surroundings for the image on screen without reflecting the surface of the screen but a Mju/Stylus doesn't have that option. So the best approach is going to be to stabilize the camera well enough to get a good shot of the environment at a slow shutter speed. From there you can reduce the brightness of the screen until it looks right in a digital photo (and wrong to your eyes) and take the shot.

If you want a picture of just the image on screen then you are making a copy stand, and a copy stand still has the camera in a tripod or on a supporting arm so you can try to get the framing right and the best sharpness.

The film isn't going to be special in either case so it might as well be slow unless you need to shoot people (who can only sit so still). In fact, in the case where you're shooting a CRT monitor then the longer you can stretch the shutter speed the better image will look, and no handheld shutter speed will look good.