A proprietary alternative that looks to be the second strong contender to Google Reader's throne at the moment is Feedly. While Feedly is not a typical web service, the company has extensions available for Google Chrome, Mozilla Firefox and Apple's Safari and applications for iOS and Android all currently available free of charge). Feedly currently uses Google's backend service, but the company has announced that it will switch the service to its own backend before Google Reader shuts down; apparently this backend has been in development for a while. The new backend is also expected to offer API compatibility to mobile applications which used Google Reader.
Feedly's own mobile applications take a more visually oriented approach to Google Reader's headline-oriented presentation. They can also present stories by relevance instead of the newest/oldest items first. The Chrome extension, on the other hand, presents a more traditional user interface that can lists stories by headlines and in chronological order. For an even more Google Reader feel, people have developed userscripts which change the way the extensions work.
Improvements that the developers have made to the application since Google announced its decision have already brought its interface more in line with Google Reader; a trend that will likely continue, judging by statements from the developers. The Feedly developers have also put together a collection of tips for the large number of users that have been migrating to their service from Google Reader.
Netvibes & The Old Reader (P, H)
Netvibes, which describes itself as a "social dashboard service", is a proprietary, hosted news reader service that can be made to look like a more traditional feed reader. At first, its widget-based approach to presenting news might seem alien for users switching from Google Reader, but like Feedly, the company has published a guide for Google Reader refugees who want to change the look of its "reader view" to something they feel more at home with. In my tests, Netvibes (once switched to reader view and configured accordingly) provided me with a speedy and pleasant reading experience, even if both NewsBlur and Feedly were somewhat more intuitive to use. Netvibes is currently free-of-charge for individuals.
The Old Reader – which started as an alternative to Google Reader when Google removed the social features of its product following the launch of Google+ – has made a commitment to scale up to the increase in its user base. The proprietary service is currently free, but the developers have indicated that they will be switching to some sort of model that includes paid accounts to finance the development. The mission of the developers is, as the name of their product would suggest, to make The Old Reader as close to Google Reader (with sharing features) as possible. In my tests, I found the product to be a reasonable approximation of Google Reader's feature set. The service, and especially OPML imports, are currently rather slow, but I expect that is due to many of the same scaling problems NewsBlur has been experiencing and will be fixed in time.
Tiny Tiny RSS (OS, SH)
All of the services so far have the benefit that they have hosted services available for users who want the convenience of a web-based reader they can reach from everywhere, but who do not want to host the application themselves. Tiny Tiny RSS does not currently have such an option, but the software is completely open source and relatively straightforward to install. The software is GPLv3-licensed, written in PHP, and stores its data in either a MySQL or a PostgreSQL database; a client for Android tablets is also available.
The interface of Tiny Tiny RSS is relatively simple, but, aside from the sharing features that applications such as NewsBlur and The Old Reader offer, it can replace most of Google Reader's functionality. Folders, sub-folders and bookmarks are all implemented as is authentication for protected feeds. Instances of Tiny Tiny RSS also support multiple users, so setups for small groups and teams are possible. If an immediate self-hosted replacement for Google Reader is the goal, Tiny Tiny RSS is probably the best option, as it can be installed in a matter of minutes on a standard web server setup that has Apache, PHP and either a MySQL or PostgreSQL database engine installed.
selfoss (OS, SH)
A very similar piece of software to Tiny Tiny RSS, and also GPLv3-licensed, is selfoss. This self-hosted reader application is also written in PHP and uses MySQL or PostgreSQL as its database; in addition, it supports SQLite. While it has a few less features than Tiny Tiny RSS and presents fewer news stories at a glance, it does have a much fancier looking user interface. In addition to pure RSS feeds, it also supports so-called "sprouts", which are essentially plugins for sources such as Twitter and several image hosting services.
Developed by Tobias Zeising, selfoss currently doesn't offer mobile applications, but it sports a responsive design that is very accessible from both mobile phones and tablets. This does, however, exclude the possibility of reading news on the go without an internet connection. The ease of install of both selfoss and Tiny Tiny RSS are comparable, the former doesn't currently support multiple user setups, however. The source code for selfoss is available on GitHub.