I have PCOS and one of the delightful (please note oozing sarcasm) symptoms is VERY sensitive skin with most toiletries. Among other things, I've made my own soap for over 20 years and have experimented with different oils and butters. However, I have not considered their sustainability until reading this thread. My focus was always on my skin comfort....which is a HUGE. However, sustainability will be in my mind going forward.
One of my favorite oils is olive oil. Usually I can get it fairly inexpensively at membership stores or grocery stores believe it or not (not the extra virgin, but from subsequent pressings). I've used pomace olive oil as well. Both work nicely...I go with what I can get cheaper. I also like coconut and almond oils. Their combo along with olive makes great soap and will give different properties depending on the ratios you use.
One thing to consider is the properties of the oils used. Not all are created equal. For example, my mom liked using Crisco for the fat for a few years. She loved the soap, but it melted too easily with water. My skin was not happy with it.
If someone is just starting out with soap making, my suggestion would be to experiment with different oils/fats (and combinations) to see what you like best. You may like the leftover fryer fat if it's mixed with something else. Also, keep in mind superfatting soap to give it more moisturizing properties. I usually superfat with cocoa butter, or shea butter.
I make an olive oil soap with freshly ground rosemary and oregano for times we need a bit more antimicrobial power. It bubbles up nicely and washes away clean. It works great and you don't smell like spaghetti as much as you would think...hehehe
For my cinnamon, oatmeal, goat milk soap, I use less coconut and more olive and almond. The soap doesn't bubble up much which bothers some people, but to me is wonderful. Less drying and the properties of the ingredients seem to work better on my skin because they aren't washed away as much. (To give an example of how powerful ingredients can be, this was also a favorite of a nurse who was helping me take care of a dying loved one as it helped my loved one's skin greatly. I loaded her up with as much as I could to use on other patients.)
Happy soaping everyone!