I've been applying some adhesive-backed veneer to some shop cabinets just to get some practice since I've never done it before. Didn't have much trouble getting it on, but trimming it has been another story. I've tried going nice and slow with a utility knife with a new blade, and then I tried one of the double-edge trimmers that you squeeze and slide along the edge. Both have had terrible results. Even when I'm not getting tear-out or splintering the resulting edge has a distinct waviness to it (ok, just the tear-out and splintering when using the trimming tool). How can I get results I can stand to look at?
You really need a router. Either a laminate router (the are small and easy to handle) or any router really. Next you need a trim bit. They have MANY different types. Flush cut, 45, 25.5, finish bit. The finish bit will leave a 1/16 laminate left to sand/file, or you could set the 45 bit to get almost flush.
Another way is with a sharp blade in your knife and a piece of wood. Take a scrap piece of ply 4" wide and how ever long you need. Place flat on the "face" of the cabinet and let about an inch hang over the edge. We are using the wood as "backer" when we score the laminate. Now take the knife and score the underside of the laminate a few times pressing the knife to the laminate and the lam to the wood, using the edge of the cabinet as your guide. Do this on a practice piece to get "a feel for it". Either cut all the way through, or score enough and just snap the lam.
* Edit * If you do use a router, apply 3/4 masking tape (or blue tape) to whatever surface the bearing will ride on. This will prevent any marks the bearing will leave, mostly a "shiny" line.
If you don't already have a router or the right bit or you want to to the hand-tool route, you might try an edge trimmer like this one on Rockler's website. It adjusts to the width of your panel and is made specifically for this purpose. They're also pretty inexpensive. You could make your own pretty easily using a razor blade or two and some scrap wood if you're so inclined.
So as to not show a preference for Rockler, please note that you can find similar items on Woodcraft, Amazon, and a variety of other places. You might even find one your local blue or orange big-box store.
To reduce tear-out, be sure that you are trimming with the grain and that your blades are super sharp. A trimmer with a skewed cutting angle can also help. And only squeeze as much as you need to to get the width you're looking for, which may be easiest if you squeeze to the desired width with one hand and push the trimmer along with your other hand. If you're squeezing hard enough that it's difficult to slide the trimmer along, chances are you're grabbing fibers and tearing them out along the way. You can also try taking multiple shallow passes and creeping up on flush. Lighter shavings are easier to control.
If you want to be extra careful, trim close to flush using any method (double edge trimmer, razor, chisel, handplane, etc.) and then sand or file the last little bit flush (as suggested by Shirlock and Bib). You can incorporate this into the normal sanding regimen for the rest of the part.
Place a steel rule along the face of the board face so it sits on the supported veneer edge ie edgeband down on a scrap board run the blade along the edge of the rule until the scrap comes away then you have a straight edge about 0.75mm over the board to sand carefully with a very flat block and paper
A single edge razor blade works well. Even then it is a good idea to leave the edge a bit high and finish the trimming with a sanding block.
An even better approach to edging is to laminate a thin strip of solid wood to the edge, an eighth or a quarter inch is good. This eliminates the unsightly area as a glued thin veneer edge inevitably wears over time.
The edge trimmers work the best when the stock (not the edging) material is delicate. Routers tend to scuff or beat up the stock where the bearing rides. To avoid tear-out, be sure to follow the grain. To do this when the grain isn't parallel with the edge, you will need to trim one edge at a time. Sometimes the grain shifts direction, as well, meaning that you'll need to trim some in one direction, then come back in the other direction to finish. This trimmer works well. You can take it apart easily to just trim one side.