This article is from the Photographic Lenses FAQ, by David Jacobson with numerous contributions by others.
Open the back and, if necessary, trick the camera into opening the shutter and stopping down. Imagine putting your eye right in the corner of the frame and looking at the diaphragm. Or course, you really can't do this, so you have to move your head and sight through the corner of the frame, trying to imagine what you would see. If you "see" the entire opening in the diaphragm and through it to object space, there is no vignetting. However, at wide apertures in most lenses the edge of the rear element or the edge of the front element or filter ring will obstruct your vision. This indicates vignetting. Try to estimate the fraction of the area of the diaphragm that is obstructed. Log base two of this fraction is the falloff in f-stops at the corner.
You can also do this from the front. With SLRs hold the camera a fair distance away with a fairly bright area behind the viewfinder hole. With non-SLRs open the back and arrange so a reasonably bright area is behind the camera. Look through the lens, and rotate the camera until you are looking right at one corner of the viewing screen or frame. (If you are using the mirror-down technique with an SLR, choose an upper corner of the frame, i.e. look from below the axis.) Now for the hard part. Look at the aperture you see. If there is vignetting you see something about the shape of an American football. If the filter is causing the vignetting, one of the edges of the football is formed by the filter ring.
A third way to detect vignetting is to aim the camera at a small bright spot surrounded by a fairly dark background. (A distant street light at night would serve well.) Deliberately defocus the image some and observe the shape of the spot, particularly in the corners. If it is round there is no vignetting. If it looks like the intersection of some arcs (i.e. like an American football), then there is vignetting. Note that near top of the image the top of the circle may get clipped a bit. This is because in many cameras some light (from the top part of the image) misses the bottom of the mirror. This affects only the viewfinder, not the film. You can use depth of field preview (if your camera has it) to determine the f-stop at which the spot becomes round. With wide-angle lenses the circle of confusion may not get large enough for this technique to be useful.