I've used the Google Reader client
Byline. app on my iPhone for over a year. The latest version is the best to date, but it's still has synchronization problems. It continues to show me articles I've read.
Reader itself has problems of this sort, but Byline is significantly worse. So lately I've been trying
Reeder. I wrote this quick review for the app store ...
Bad news first. It crashes more than it should, but not more than Byline. Secondly it needs a manual, and it desperately needs a "reset" button to wipe its local store and force a reload from Google Reader. Thirdly it get its sync state confused, but no more than Byline. Lastly it doesn't precache the source pages, so Byline has a big edge there. There's no support for creating a Google Reader "Note" status update (Byline doesn't either).
Now the good news. The under-documented UI is elegant -- once you figure it out. (Programmers raised on games think life is a role playing adventure.) Readability is excellent. There's an option to open source pages in Instapaper Mobilizer - a vast improvement over Google Mobilizer. Services and configurability is excellent. Performance is great, so stability is now a bigger issue. It shows Following (Byline doesn't) - but here it gets counts wrong.
Bottom line - definitely worth the money, currently the best of breed, room for stability and synchronization improvements and, for the love of Binary, please add a reset option.
I think Reeder is a better app than Byline -- for the moment. It's not perfect though. If the developer doesn't fix a few bugs soon I'll take a look at
Mobile RSS next.
See also (reviews found by search on [Reeder Byline "Mobile RSS"]:
Update 9/30/10: No contest. Reeder is much better. There are bugs (Followed count), and crashes, and I miss the original page pre-caching -- but it's still the clear winner.