Here's a slapped-together list of RSS feed readers, posted ahead of Google Reader's impending shutdown on July 1:
Feedspot -- A correspondent described this as "the least bad right now."
Feedly -- Particularly wired to import Google Reader data - do it soon. You still have to login through Google - still waiting to see how they handle logins after 7/1/13. Update, 9/22/14: Feedly is still forcing you to sign-in with Google. See Yves Smith on the yuck factor of "social logins." Update, 1/21/2016: Feedly finally added email logins so you don't have to use Google anymore to access the product.
Bloglines -- I've used this for years. They seem to have switched recently to requiring a login every time you view (i.e., the "remember me" button is worthless). Update, 9/22/14: Bloglines is still going, under the umbrella of something called "netvibes." You stay logged in, but the site has a penchant to go offline for 1-3 days with no warning or explanation. Update, 1/8/15: Bloglines users had their feeds moved to the Netvibes platform sometime last year, without any explanation of the switch. My Bloglines login continued to work on Netvibes but last week the Bloglines "dashboard item" stopped working. I am continuing to use Netvibes as an alternative to Feedly, but I had to manually re-add the feed URLs. Netvibes is a subsidiary of a French company, Dassault Systèmes.
The Old Reader -- Modeled on Google Reader before it was merged into the Google Plus empire of hell.
Livedoor -- From Japan -- I get incoming links from this.
Bonus: Hundreds of tools unironically wishing Google Reader a happy retirement. All of these comments should say, "thanks for nothing, too bad your attempt at web hegemony failed in favor of another attempt at web hegemony, yours truly, victim"
Update: Go Read explained in a blog post (hat tip Pretzel). "Open source" but Google login required.
Update 2: Naked Capitalism recommends Digg's new feed reader, particularly as an alternative to Feedly and its design in "hipster Brooklyn green." Update, 9/22/14: See Naked Capitalism's Yves Smith on the yuck factor of "social logins," including Digg's.
Update 3: AOL Reader (still in beta) - you have to sign in using AOL's or one of the other big evil spy companies' logins, natch.
Update 4: Layout for tommoody.us as seen on bloglovin'