The .museum domain has begun its startup procedure. The startup rules can be seen in the ICANN website. The procedure is a nostalgia-inducing throwback to the "good old days" of the "academic and geek Internet" that existed when the original top level domains were introduced in the '80s -- rather than having a complex "sunrise" involving trademark lawyers, corporate marketing types, and domain speculators, they're establishing the naming structure, and taking requests for the museums' names in this structure, through a process of open discussion and consensus.
Representatives of museums (limited to not-for-profit institutions with exhibits open to the public) are able to discuss the structure of the domain (where individual museums will only be able to register at the third or higher level, beneath subdomains representing types of museums, like art.museum) in an open mailing list, and request by email the name they want -- during the introductory phase, the structure will be experimented with and names assigned temporarily (subject to change). By the end of the phase, in a few months, the structure should be established and permanent registrations can begin.
This system may actually work in such a small and cooperative group as museum operators, and looks like it will produce a smooth TLD launch free of the massive problems that the larger-scale ones like .biz and .info are having. However, due to the niche community it targets and the lengthiness of the three-level names that will be used, the .museum TLD might not make that much of an impact on the public anyway -- they probably won't be frequently-used and memorable enough to make a dent in the narrow-minded perception that domain names are a flat namespace ending in .com.
There could, however, be a cumulative effect if lots of these "niche" TLDs are added; eventually, if you put together enough subcultures, you've got a sizable chunk of the user base, so people would slowly get used to the existence of other domain endings, and maybe find it refreshing that there's less hype and fewer marketing gimmicks in them.