1. Fiskar kids scissors (blunt tips, sharp blades) and a thread nipper are all you really need and you have to be careful with both.
2. On smaller mats, use the scissors to cut the top off horizontally and you'll see two to five "columns" of mats. You can carefully cut down the very light mats between them, and then it's easy to cut the columns fairly close to the skin. A week or so later, they will have grown out enough to cut and have only fur left.
3. On the bigger mats, where they're attached to fur all the way around, carefully use the nippers on the edge, keeping them horizontal to the skin, and you'll see the mat is open under that part. Keep carefully going around cutting the parts that connect the mat to the fur, and then work like #4.
4. The best way to deal with the bigger mats once you have some open edges is to make them smaller. Use the nippers carefully to work through a part of the mat (I have a quarter of that giant mat on Loki off now) and once it's separate, you can use the scissors to gradually cut it down. Eventually you'll have fur at the bottom and you can cut there and get the entire mat part off. Continue for the rest of the big mat.
For some reason, when you make the extension of the mat smaller, the bottom of the mat doesn't mat right away. The fur grows out a bit and that lets you cut the mat clear.