I obviously have an affinity for BeyondPod, which I think has a great interface and options to discover new podcasts that you may not have known you'd be interested in. The real selling point for me though was the ability to add podcasts to the BP playlist or have the app choose which podcasts it thinks you might want to listen to. BP uses its own player, so you can reorder the list when you want, pause, and skip through tracks.
My only issue that I started having was that one of my podcasts stopped the audio play, which was odd as it had previously been able to play both video and audio. Since I listen to these as I go to bed, the audio is much more important.
I am currently using G Reader just for a change up of the apps on my phone and I find it to be adequate for what I want to do and that's listen to podcasts. RSS Demon is also a good choice, but I found it to be quirky at best, though it could also be my phone and not the app.
Feedly was another okay app, especially if, like me, you use the website version on your computer -- the downside I had was that I wasn't able to play podcasts. The app is still in beta though, so hopefully the official app will include podcast play, especially when it's a standard feature of the web version.
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