I've been doing some experimenting, and agree that Dove works the best. Problem is, there's about 15 varieties of Dove soap, including a "calming" one (but, oddly enough, none for shaving). I chose the one for sensitive skin.
The key advantage of Dove is its neutral pH. No stinging, thick lather.
Yardley is too alkaline, as are most hand soaps. And that Yardley with oatmeal really does have oatmeal in it, which I do not want in my shaving brush. But it smells (almond) and washes really nicely -- excellent bath soap.
Dial white seems a lower priced version of Dove, and is worth trying. Their red Power Berries semi-glycerin soap is also very mild and nice.
Stay as far away from Kirks as possible, unless you want to use it as a laundry soap. WAY too alkaline.
I also found a common glycerin bar soap that works very nicely.
Frankly, I think these soaps work a lot better than many shave soaps.
If you are experimenting, start with Dove. I'm still experimenting, myself.
Dove does contain some soap, but the primary ingredient is a detergent. That is how is is more neutral in pH. Here are the ingredients:
Sodium Lauroyl Isethionate, Stearic Acid, Lauric Acid, Sodium Tallowate or Sodium Palmitate, Water (Aqua), Sodium Isethionate, Sodium Stearate, Cocamidopropyl Betaine, Sodium Cocoate or Sodium Palm Kernelate, Glycerin, Sodium Chloride, Tetrasodium Etidronate, Tetrasodium EDTA, Titanium Dioxide (CI 77891).
Sodium tallowate, sodium palmitate, sodium stearate, sodium cocoate and sodium palm kernelate are all saponified fatty acids (soaps). If you notice, the ingredients say they can use sodium tallowate or sodium palmitate. They also say they can use sodium cocoate or sodium palm kernalate. If the soap contains sodium palmitate and sodium palm kernalate, it will perform much differently than if it contains sodium tallowate and sodium cocoate. I try to avoid products that indicate they will use whichever ingredients are more available or less expensive at the time the soap is manufacturer. Different ingredients means different performance. That may not be important when washing your face, hands or body, but it is important for shaving. Thus, you next bar of Dove soap may not perform like your current one.
Sodium Lauroyl Isethionate, Sodium isethionate and Cocamidopropyl Betaine are synthetic detergents. I do not mind detergents in my shampoo and hand soaps, but I try to avoid them in shaving soaps.