There are tens if not hundreds of MSR designs. Here's a useful report published by the International Atomic Energy Agency on the status of MSR in the world [1].
There are two types of reactors that hold the promise to burn a higher fraction of the fuel: thorium based thermal reactors and uranium fast reactors. Many such reactors are MSRs, but so are other reactors. Honestly, you can think of MSR as an orthogonal dimension.
In any case, to answer your question about which designs moved forward. You can think of a design moving forward if the company that promotes that design has applied for NRC approval, and how far they are in the approval process. Here's the page where you can find these details [2]. There are 12 companies listed there. Any north-american company that is not listed there, you can think of their design as "pie-in-the-sky". Among the 12 companies, there are 3 molten salt reactor designs. One is in a reasonably advanced stage (the Kairos-Hermes), but that is a classical MSR: it uses uranium and thermal neutrons, so there are no ambitions to breed new fuel or anything. That's good, you want a design to be as conservative as possible if it were to get an NRC approval. The Terrestrial Energy design is also thermal, the only fast MSR is Terrapower's Molten Chloride Fast Reactor design. However, Terrapower is in a much more advanced stage with its Natrium reactor, its MSR is basically on the back burner. My guess is we won't hear any news about this design for at least 10 years.
However, the Natrium design of Terrapower has some promise, and quite solid funding. It is a sodium-cooled fast reactor. If it succeeds, such designs can burn U-238, and that is somewhat of a game-changer.
https://www.funraniumlabs.com/2024/04/choose-your-own-radiat...