The obvious answer is MWF, but I also recently discovered that Williams (tallow) is a completely different soap with soft water. We have had work done in our cellar and we disconnected our water softener. Yesterday's shave with Williams was a bust. Probably the worst I have ever had. (You all know that I am a Vintage Williams Junkie). This morning, before shaving, I reinstalled the water softener. Low and behold, my beloved Williams came through with flying colors. Any other soaps in your arsenal that are finicky enough that soft water is a must?
In my experience, all soaps I've used, both tallow and vegetable based, are easier to get going with softer water. I have very hard water (17 g/pg where 10 is considered very hard) and the only soaps I've had difficulty with are a couple of vegetable based soaps. I was able to get excellent results with MFW and other soaps considered difficult with hard water--it just takes a little more work, which is expected with my water. When I travel to areas with softer water, lathering is easier and, while the consistency of the lather may be slightly different, it's not noticeably better. To answer your question, I guess only a couple of soaps I've used over the years that I would say soft water is a must (for me at least), but most of those formulations have changed for the better (Classic Shaving early formulation is one). You may need to change your approach and work a bit more, but you can get there with any of the modern soaps I've tried. @Mantic59 did an excellent review of soaps that work better with hard water. https://sharpologist.com/2017/01/best-shave-product-hard-water.html
I get good lather with vintage or modern Williams with whatever water I use. Softer water is easier but hard water works good aswell. I bet if you had a few more shaves on the hard water it would sort out.
When I say hard water, I am not exaggerating. I live just west of the #1 city for hard water in the nation.
You must be in Colorado? A member from Colorado has water in excess of 20 g/pg, which you can walk on. He must slice off a piece of water to add to his lather bowl.
#1 Indianapolis (see attachment) And from what I am told, the outlying regions are worse than the city, which does something or another to the water. Haven't walked on it yet...nyuk, nyuk, but I feel my bones hardening daily... I am prone to Kidney Stones, so a softener is a must! Maybe I'll start a Kidney Stone thread.
I've got hard water. Normally, a day off or vacation time means no shaving for me- just let it grow. I went to a wedding out of town a few months ago and shaved at my in-law's place. Super soft water and MY GOD WHAT A DIFFERENCE! I've never really thought about a soap not working well with my water, I just thought it was a junk soap. If you have soft water your slickness is probably 25% better than mine. So, to answer your question I guess all soaps would benefit.
Good ol' Indiana limestone. I used to live on the far east side, and the water there was crazy hard. If you fill a pan with water to soak before you wash it, don't dare forget about it. In the morning there will be a white ring that has to be chiseled off. I've since moved down to Beech Grove, and we just don't have that problem here. I don't know if it's treatment, or a different source, or what. But I sure don't miss it.
We installed a softener that changes the charge of the minerals so they don't stick to stuff quite so bad. It's not so great and doesn't help with a shave at all. I'd go with a salt system if I were to do it over again.