Stupid question. What makes soap biodegradable?
All soap is effectively biodegradable. It's the levels of nitrogen, phosphorus and sodium present in soaps and shampoos that can make them hazardous to your garden.
(There's also the difference between a soap and a detergent, but I'll leave that one to the Bronners of the world.)
It's not as big an issue with shower water since your bath products are liquid and so highly diluted by the time they hit the bucket, but powdered washing machine detergent can be comprised of more than 3/4 of a filler which is often really high in phosphorus and sodium.
Avoid collecting water when and if you use dandruff shampoo. Also avoid laundry products like Borax which contain boron, as it kills plants in high concentrations.
There are a ton of sites out there which tell you whether or not a particular product is greywater safe, but you can always check the manufacturer's site or email them if in doubt.