Soaps made with coconut oil and minimal additives
After learning that soaps could alter endocrine function, I started paying more attention to their ingredients. But I only recently learned about two soap brands with extremely basic (I can't promise that's not a PH joke) ingredients:
Liquid coconut oil soap
NutriBiotic makes a nice liquid soap that contains nothing but water, coconut oil (virgin, unrefined, non-GMO, saponified with alkali), and citric acid. This is runnier than your typical liquid soap, and it won't create as much pleasing foam because of the lack of foaming agents. But it seems to work well as both a hand and body wash. It has no noticeable smell, unlike the next option.
I'd suggest getting a pump bottle for it, it's runny enough that the pop-top bottles aren't ideal. I bought a couple of these bottles, and they work fine for dispensing it.
I tried it as a shampoo, and it worked fine. However, I measure it at a PH of about 8.5, so it might be a bad idea to use frequently for that purpose.
Bar coconut oil soap
I learned about this brand thanks to the ever-spotless Danny Roddy. It's nothing but coconut oil and lye. Unlike NutriBiotic, this one smells strongly of coconut, and on the expensive side. But I grumbled to myself and ordered the 28 pack to reduce the unit price, and have been happy with it.
It foams up well, and works as a shaving soap in a pinch.
Although the soap itself has a strong smell, my professional Dan-smeller confirmed that it didn't leave me smelling like coconuts. Whether that's a good thing is up for debate.
The same company also makes coconut oil laundry products. I haven't tried it though.
Homemade coconut oil soap
I haven't tried making my own soap, but here's two useful guides for doing so:
I think they're both worth reading before making an attempt because they have complementary information.
Choosing common soap brands in local stores
I haven't looked into most common brands of soap, but some of them will use coconut oil as a source under names like sodium cocoate. I'd suggest looking for that, or other saturated options like sodium tallowate (unless you wish to avoid animal products) and sodium palm kernelate.
For additional ingredient checks, The Environmental Working Group's has a useful database of skincare products and their ingredients. Their product search will return product rating pages, but also can be used to find a "hazard score" if you search for an individual ingredient.
My previous default easy-to-find soap brand has been Ivory. In spite of the technical-sounding ingredient names, they should be mostly safe. The only ingredient I'm not sure about is the fragrance: they're only able to tell me that it's "a combination of organic and synthetic ingredients." However, you'll be relieved to hear that they have their priorities straight: by saving time not reporting full ingredients, they've been able to create an official site full of attempts to spread viral soap memes. Sigh.