I trim the front side of the quilt after it's washed. Lots of those pesky threads come out in the laundry. I don't trim the back of the top very aggressively. I try to remove any dark threads that might show up under light fabrics, but I don't obsess over them very much.
Another thing I do. I don't actually "cut" the threads. I bring the scissors down to the fabric, put some tension on the thread and just push the V of the scissors blades into the thread. That cuts the thread without risking cutting the quilt, because you never actually move the blades across one another. I read too many horror stories of quilters snipping holes in their quilts while trimming threads. I decided it would never happen to me.