The "Rat's Nest"
You can use those small offcuts and trimmings of fabric tucked under the point of the start of the seam as Vanessa at Crafty Gemini advises, but that leaves you with a couple of issues, you can then cut the fabric away and leave yourself with a perfect point, there is a danger of cutting into the stitches themselves, and no matter how close you trim you have an extra layer of fabric under the thread which can become an issue depending on how your seams are layered.
Easier and more efficient is to use those trimmings of paper from your paper piecing blocks. You are probably like me, in that you save the bigger bits and staple them together to use as scrap pads and memo pads, "waste not, want not". Some of the trimmings are just too small to be useful though, these are the bits I save next to my machine, When you have a sharp point that the machine will chew up, or just very fine fabric that causes those threads to bunch underneath, no matter how careful you are, it will happen, even the BIG names in the sewing blogosphere have the problem. Put one of those skinny bits of paper underneath though and it does the job beautifully. The feed dogs have something to grab on to so the fabric feeds through smoothly, the threads are separated to they can't twist up, and when you are finished, you can simply tear the paper down the perforations made by the needle. Go carefully if you have delicate, or sharply pointed starts to the seams, you don't want to ruin the stitching by distorting it, bend it backwards and forwards a few times to make the perforations easier to split. If you have really fine fabric say Organza or Chiffon, hold on to the starting/ending threads while you pull apart the perforations, Although that might seem like a lot of information therefore a lot of work, still, it is in all ways easier to use paper. No more tangled threads or chewed up corners, and all with bits you would have thrown away anyway.