Google’s Think Insights has some free research on The World Has Gone Multiscreen: How The Mobile Internet Changes Our Lives. The key takeaway is that multi-screen is the new normal. The global average is nearly 2 connected devices per person. The majority will soon own smartphones and tablets are rapidly catching up.
I believe too few companies/developers think about multi-screen when designing apps. Multi-screen is much more than data sync and functional and UI replication. If budgets allow, avoid the lowest common denominator. Instead, think about designing the app for typical use on that device and retaining each device look and feel. In some cases syncing settings can be more useful than syncing data. Consider catering for multiple users on particular types of device. For some apps, explore funnelling your users onto the device where you are most likely to make a sale. Also, there can be opportunities to control one device via another.