I don't have as much of a problem when sewing/hemming sheers & lightweight fabrics, but when I cut them I can never get clean edges. I read a post earlier about not cutting on the fold. That sounds like it might be a good fix, but then I'd have to transfer the pattern to paper and draw the whole piece. I don't really have a problem with that, but I thought others might have some tips?
Have you tried using tissue paper under your sheer fabric? Sandwich the fabric between the tissue paper and the pattern piece. This provides some additional stability. Also, try using a rotary cutter where you can.