Here are a few of the most popular Reddit clients on Android and iOS:

Android

• BaconReader: This robust mobile app offers a great UI and all the functionality of the full site. Drill down into the comments and bring up options on any thread by swiping left or right. Our one gripe is the inability to share Reddit URLs via other apps (Gmail, Twitter, etc.). However, you can easily share the content. BaconReader is ad supported, but you can pay $1.99 for the ad-free premium version.
• Reddit Is Fun: Thanks to some recent updates, this app has come into its own in terms of user experience. It offers rich functionality as above, a "Dark Theme" for nighttime browsing, and the ability to view any thread in your phone's browser. You can grab the ad-free premium version here.

iOS

• iReddit: Reddit offers its own really sleek iPhone app that lets you swipe through your customized homepage and view content without leaving the app's environment. It also functions nicely on the iPad.
• Alien Blue: Alien Blue for iOS is a fluid, organized Reddit viewing tool for your iPhone, iPad or iPod Touch. The app supports multiple accounts, has a night-mode and is updated regularly.