App Icon not showing up, although I have added in Xcode 5
This might be an XCode 5 bug that the added icons are not actually included in the app bundle, but you can work around it by going to Build Phases, expand Copy Bundle Resources, then press the "+" sign in the bottom to manually add the icon files to this category.
In addition iOS 5/6 seems to have a bug that the screen doesn't update the app icon even if the icon files are included in the bundle. To work around it (as suggested by josema.vitaminew at App Icon not changing when app version is updated in iOS 5 simulator) you can drag the app icon into a folder, then iOS will update the appearance of it.
Expanding on adp's answer, if anyone has switched to asset catalogs but they stopped working, and still can't find a solution for Asset Catalog, Images.xcassets, AppIcon or LaunchImage not working, here is a reproduction of the same problem I encountered, and the solution:
I had upgraded to Xcode 6 and opened a project created by an earlier version (4 or 5, not sure) and no matter what I did, AppIcons and LaunchImages absolutely would show up when I ran the app in either iOS Simulator or the device.
I tried Xcode->Product->Clean, Xcode->Window->Organizer->Projects->Derived Data->Delete..., relaunching Xcode and the iOS Simulator, then rebuilding and relaunching the app (failed).
I tried "Reset Contents and Settings..." in iOS Simulator and even deleting ~/Library/Developer/Xcode/DerivedData//Build/Products/Debug-iphoneos and relaunching the app (failed).
I tried archiving the project, hoping that it was just an issue with the iOS Simulator, but the icon in the Organizer-Archives window was still shown as the default white square with lines, confirming that the icon would also not show on a device (a helpful trick to have when no device is handy). (failed).
I tried choosing "Don't use asset catalogs" in Target Settings and creating a new AppIcon-2 and LaunchImage-2 (failed). -this is what cost me at least 2 more hours, because this should have worked and the fact that it didn't revealed the bug in Xcode but I just couldn't see it-
I tried creating a new project from the single view template and compared all of the Info.plist settings (failed).
I tried comparing the new project's Contents.json with mine in Images.xcassets (failed).
I tried dragging the images from the left sidebar Navigator into the AppIcon image wells, hoping that they were just not being copied in a build phase for some reason (failed).
Finally in exasperation at the thought of having to create a new project just to fix the icons, I tried deleting Images.xcassets in the left sidebar Navigator and choosing "Move to Trash", then quitting Xcode and relaunching in case it cached anything. Then when I went to click the App Icons Source popup menu, it only showed "Don't use asset catalogs". I had to select it in order to bring back the "Use Asset Catalog" button. Then I was able to create new AppIcons and LaunchImages, drag images to the wells, build and launch the app in iOS Simulator and finally see icons appear properly.
So the problem appears to be with how Xcode stores Images.xcassets inside the project, and not with target membership, derived data, or anything in Images.xcassets itself. You must remove your assets file and create a fresh one, then drag images to wells by hand again.
It took me 3 hours of research to solve something that should have "just worked" and the entire process of having to manually create individually sized art assets has been so tedious over the years that it has personally cost me countless hours of busywork.
Hopefully this bug workaround helps someone avoid the frustration I went through. I can't help with the friction of manually sizing art assets, but I highly recommend the Icon Slayer website to automate that (of which I have no affiliation).