Regular liquid laundry detergent. The only bike-specific clothing I have are shorts. I typically hand wash in a basin of cold water/soap (also throw the gloves in the mix); knead them, let them soak for 10 minutes, then knead again. Rinse, then hang to dry in the backyard.
FWIW: I've also used shampoo for washing, too. I had some off-brand unscented baby shampoo that I never used, so rather than throw it out I used it for washing shorts. Worked fine, and the bottle lasted about a year.