I don't know the breakdown between storage types. Personally, I think the other types mentioned, pumped storage hydro and underground heat storage, both of which are used in large scale - especially pumped storage hydro, are going to be used more than fuel cells.
https://en.wikipedia.org/wiki/List_o...orage_projects
Pretty much all the large existing energy storage facilities are pumped storage hydro power. The nice thing about this is they are currently needed for exactly the opposite times as they will be needed when solar energy gets to the point that storage is an issue. Currently they pump during the night because there's excess demand during the day. At some point they will pump during the day for demand at night.
I think other large scale storage facilities will be more similar things like pumped air pressure in abandoned mines rather than anything high-tech.
But, as far as batteries go, I think there will be a lot more small on-site energy storage, greatly reducing the need for utility storage.
I'm not completely anti-nuke. It does threaten the environment. Accidents happen. They are terrible. There is waste that lasts forever. Probably a couple of those fancy new reactors that process old waste would be a good idea.
The big problem with nuclear is it costs more than it seems because subsidies are less direct and don't count externalities. Construction takes forever and for the most part it will be obsolete before any new reactors can be built. The newest reactor in the US took 23 years to complete - that was phase 1, phase 2 started 42 years after construction started.