Feedbin is a simple paid web-based RSS reader with a well-supported API.
Ranked in these QuestionsQuestion Ranking
Pro Clean UI
Uncluttered and intuitive to use.
Pro Classy interface
The web interface is nice, and the apps give a great native experience regardless of what platform you're on.
Pro Full API
Feedbin has a fully-featured, free, REST-style API.
Pro Free and open source
This won't disappear in a year or two if ad revenue falls off.
Pro Supports JSON Feed
JSON Feed is a new syndication format that uses JSON instead of XML to store its data. JSON is generally considered easier to read and write than XML, making for less error-prone data structuring and broken feeds.
Con There is no free plan
One of the few news readers that does not offer a free plan. They have a 14 day trial, after that a subscription is $3/month.