The most common compatibility issues arose when I tried running apps that I had synced from my Android devices to either notebook. Instagram refused to work on them (which I found personally disappointing since there is no Windows or web version of this very popular social-sharing image service). The Gmail and Google Drive apps I synced wouldn't either. For that matter, I synced the Google Play Store to both notebooks on the off chance that maybe this would let me access the official Android app store through BlueStacks. Nada. Surprisingly, the Google Voice app worked on both notebooks. So just because an app is officially made by Google, and uses its services, doesn't necessarily mean it will work on BlueStacks.
Most of the synced apps I did get to run successfully on BlueStacks, including eBuddy, were text-based in which graphics aren't significant in their overall design and use.
BlueStacks doesn't provide a way (at least obviously within its application menu settings) to "side load" apps. Many Android devices will let you directly install apps in their native file format (they come packaged with the .APK extension).
Lack of touchscreen
Because you're using a mouse or notebook touchpad to interact with these Android apps, which were originally designed to be used with a touchscreen, you can't pinch-and-zoom on the screen of an app that provides this or other interactivity that requires multi-point touch. If you're using a mouse with a scroll wheel, then the scroll wheel works as a substitute for pinch-and-zoom.
A few early reviews of BlueStacks when it was first released toward the end of March 2012 reported that it wouldn't completely uninstall if you wanted to remove it. In my experience, fortunately, I was able to easily uninstall the current Beta-1 from both of the Windows notebooks on which I tested it.
Conclusion: What's the point?
So why bother with using BlueStacks if you already own an Android device? Perhaps you'd like to occasionally use your apps on a traditional computer for its full-size keyboard and larger screen. Or if you don't own an Android device, BlueStacks lets you enjoy apps that have no Windows or web equivalents (though not Instagram, sadly). If you are an Android developer, you might be able to use BlueStacks to test your app. Compared to the mobile OS' official emulator that's meant for development use, BlueStacks runs much faster.
If you're hoping to use many of your favorite Android apps on your Windows PC, especially those that you downloaded from the Google Play Store (and these apps make extensive use of graphics or use Google's services), the odds are that they won't work under this current iteration of BlueStacks. Basically, compatibility with synced apps that you originally downloaded from the Google Play Store is iffy in the BlueStacks Beta-1. You may be better off trying only apps that are offered through the three app stores (1Mobile, Amazon and Getjar) that come with BlueStacks.
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