What Google's Algorithm Thinks of the New Top-Level Domains
Google's official position is that the new domains do not improve or hurt search rankings. Google has long sought to reduce the importance of domain keywords due to a combination of spammers squatting on keyword-rich domains and newer, quality sites having difficulty finding available domains.
However, it's still possible that behind the scenes, the domains are used as a small factor in determining the relevance of the site.
Are There Other SEO Benefits to the New Top-Level Domains?
There are possible soft benefits to the new domains. One is in click-through rates. A catchy name might be more likely to attract a searcher's attention than a generic .com.
Another is in gaining links and social shares. Again, a catchy name could make someone more likely to share your site. A specialty name may also signal that you yourself are a specialist and make someone more likely to choose your site as the one they want to link to.
What About Local Search?
At this time, Google says it is not using localized top-level domains, such as .nyc or .london, as a way of determining a site's location for the purposes of local search. This is because other top-level domains that were intended to be local became popular for other reasons. For example, most websites using .tv domains relate to television and entertainment and not the country of Tuvalu that .tv was created for.
Google will watch how the new domains are used to see if it should give them weight in the future.
What Happens if You Switch Domains?
If you decide you like the new top-level domains more than your current domain, you can make the switch without impacting your SEO. You can set up redirects to your new domain similarly to how you change your physical address with the post office. This makes sure you don't lose credit for the links to your old domain.
In the end, you can do whatever is right for your marketing plan. If you have a good .com, it may be best to stick with what's working. If your current domain is too long or just not the right fit, switching to a new domain could help boost your site.