I tried butchering some unprocessed beef for the 1st time today and had a hard time removing the tough fat & silver skin. The meat pieces were about 40-80% the size of my fist, which is way smaller than examples in the videos I've found, and they had fallen apart quite a bit, so I didn't have a large whole piece I could just slide the tough stuff off of. I used a knife the size of a pairing knife but curves kinda like a filleting knife. My biggest issue was gripping the stuff. I'm not sure what was the fat & what was the silver skin, but most of it didn't peel off easily like in the tutorials. I started each trim by piercing the fat/silver with the tip of my knife, then trying to grip a piece of cut-off fat/silver, which was very difficult as it just kept slipping out of my fingers. I then tried to slide my knife under the fat/silver, which was also difficult because I couldn't get a grip to pull on the fat/skin. How can I trim the fat & silver skin off of small pieces of meat, considering it's too slippery to get a proper grip? 
Cutting Skills  How can I trim the fat & silver skin off of small pieces of meat  place the meet on a cutting board laying it on its fascia. Hold it with your hands or a fork, cut from the middle of the piece downward to the fascia, but without hitting it and then slice just above the fascia to separate. Turn the piece around, hold the fascia you just cleaned off and repeat the slice. Any sharp kitchen knife will do the job.  Selection of the proper cut taking from your comment:  It's for stir-fry, where even the fat often gets really tough to chew through  I think the answer to this question should also reflect on selecting quality beef meat for the cooking purpose you have in mind (stir fry) rather than just the cutting technique itself. you'll need to choose from the best cuts, ideal for stir-frying:  Flank steak  Top Sirloin Steak  Outside Skirt (Plate)  Flat iron steak (Chuck)  Inside Skirt (Plate)  Beef chuck steak   keep in mind that certain beef cuts are only for stewing, boiling or roasting, which are all long term processes tendering the meat. You can cut away as much of fat or connective tissue (like fascias or tendons) of an unsuitable cut as you want, the meat will never become tender in a short stir fry. cutting knifes - my local butcher uses basically only two knifes, one is similar to your standard kitchen knife, the other one more like an descendant from an axe (to get through bones). 