I personally don't use chlorine bleaches. They are not safe to use. Chlorine bleach isn't used in commercial laundries either. The soap your shirt laundry uses most likely has an optical brightener in it that makes the clothes appear whiter. Some laundries use a technique called "blueing" where a little blue dye is added to the white loads. It is verry old school and is not common anymore, but a little blue dye makes whites look whiter. Most commonly if a commercial laundry uses a bleach they use an oxygen bleach. Most often a really strong hydrogen peroxide solution is used. (And if you get it on your hands it really burns!) The oxygen bleach is hooked up to a injector pump that automatically adds the proper amount at just the right time in the wash cycle. -Jay
Chlorine will also treat dyes the same way. In the Soviet army they used chlorine solution to mark uniforms on the inside with the names of the person. The green uniform would turn almost white in the places where chlorine touched it, and if one used a little too much it would practically burn a hole.
I use the big jug of Costco detergent. Highly rated from Consumer Reports, and very inexpensive, especially since I have an HE washer.
It will burn holes in the fabric. I remember once a restaurant worker took their white aprons in to be cleaned. They "pre-treated" all the stains with bleach. After we washed the aprons there were holes in the aprons everywhere there was bleach. The person then tried to say it was my fault... I don't think so, that's one of the reasons I don't have chlorine bleach in my store.