Authenticated DH, what protocols are secure? I read about STS + variants being insecure in the SIGMA paper, which then proposes SIGMA as a replacement. Are the SIGMA variants still considered secure or are there some other protocol that's currently recommended to use?
There have been a large number of so-called authenticated key exchange (AKE) protocol proposals in the literature since the sigma protocols that could be used to replace them. I know of too many to list. They offer various advantages over the sigma protocols, ranging from various stronger security properties (some interesting ones, some which might be of academic interest only, but certainly more than just various forms of deniability) to increased efficiency.

A good high-end design is (H)MQV, see e.g. Krawczyk's papers in 2005 (Hugo Krawczyk,  HMQV: A High-Performance Secure Diffie-Hellman Protocol, pp. 546 - 566, CRYPTO 2005.) if you are just interested in the design. There certainly was an associated patent, but I don't know what the current status of that patent is; and certainly some of its many later variants may be outside the patent.

I could mention a large number of later protocols, but they are hard to compare to each other and the landscape is quite confused; they end up being mostly incomparable if you look closely enough. I'm confident that at the moment, there is no objective method for comparing such protocols that everybody agrees on.

I'm afraid that the first person that mentions any of the newer ones will effectively start a flame war :) I'd say we'd have to wait until the dust settles down, but I think most people agree that HMQV is a pretty solid design.