My white shoelaces were dirty but not worn out yet. I hand washed them with soap and water and then painted them with white acrylic paint. When they dried, I had new laces. My laces were polyester and, if I would have used bleach, I am afraid they would have gotten less stable and rotted easily. The shoelaces will be a little stiff at first, but they loosen up. The tied bow doesn't slip now either because the paint adds stability.
I toss shoe laces into the wash with the white clothes (undies, etc) that DO get a bleach. Or you can just take a kitchen bowl, pour some bleach, hot water, let them soak 15 minutes and it is done. Nothing is going to happen to the shoelaces in the bleach. Except clean them. OR take soft scrub with bleach, wearing gloves, take laces and a squirt of soft scrub, rub like cleaning your hands, and rinse. Nothing to it.