I bought a vinyl cutter to cut fabric. I purchased a Silhouette, Portrait 3, it works, but it wasn’t the best choice. The newer Cameo line has the ability to utilize the rotary cutting blade.
If you have a machine, not capable of a rotary blade, there are some things that make cutting with a plunge and drag blade work for fabric. First, the stiffer your fabric is, no matter the blade type, the better it cuts and the less it frays. An iron on stabilizer like Heat and Bond, may be your choice, especially if you are doing no sew appliqué. Heavy Starch or a product like Terial Magic wash out after sewing by hand or machine, to the background fabric. I had never heard of this product, but it receives rave reviews for giving fabric a stiff, paper-like texture that cuts like butter and then returns to it’s fabric hand after the product is washed away. (yes, it has to be washed away)
To give the fabric the best chance at a clean cut, make sure it is well adhered. A new mat will probably do this easily, but a temporary adhesive like Scotch Repositionable Spray Adhesive or your favorite sewing temporary, spray adhesive. A wallpaper brayer or spatula might be helpful in laying the fabric on to the mat. Don’t get too hung up on specific products. The goal is a stiff piece of fabric that is well attached to the gridded cutting mat for your machine.