Which is the best detergent to use if you are going to use the waste water in your garden ?

Fairy Liquid is your best bet too clean most things, it will be kind to your garden, it is what the sea bird protection agencies use to clean the birds when an oil tanker has spilt its load in the sea, so if it is safe to use on birds and animals is will be safe in your garden. Also if you through your soapy washing up water over your rose bushes it helps kill the flies that attack them. Good luck and happy gardening.

It's not so much which is best, as staying away from those with ingredients that will harm your garden.

The most common soap and detergent ingredients to avoid are sodium, chlorine, and boron. These are either immediately harmful to plants or will become so over time, as they accumulate in the soil.