Well, I have a Singer Heavy Duty sewing machine, and I've been able to sew Layers of things together (leather, vinyl, etc.) It's never let me down, and I know it deals with fur very easily as well~
My suggestion is to never buy a sewing machine that costs less then $250, otherwise you risk having it break down on you really quick. Singers re good, Brother's are even better though more pricey. So I say try a Singer~