Silk is renowned for being difficult to sew but over my 34 years of experience painting and sewing it, I have a few ideas that might help. I have tried many of the tips given by others such as sewing through tissue paper and with silk thread but there are only one or two suggestions that I have found helpful I will share with you.
Use a sharp needle. I have bought specially fine sharp ones but really, a normal one is fine as long as it IS sharp and either not been used for heavier fabrics or a really new one.
The most useful thing I have found, that I have discovered myself by trial and error, is to control the tension of the stitching yourself rather than letting the sewing machine take the silk in itself. I hold my piece of silk in front and behind the foot, keeping it taut.
I use polycotton thread, nothing special, just what I pick up. I do have a good sewing machine (a Bernina) but it is old and has been pretty hammered. I do have it serviced (Bambers in Eccles UK) but not as often as I should!
Joanne Eddon www.joanneeddon.co.uk