Weatherbug Elite just got a really nice makeover. I have used MyCast for their radar and I always use Weather Alert USA for push alerts directly from the NWS. Although, I hear they now charge a decent price for Wx Alert - I got in when it was only .99
The thing I hate about weatherbug is the local stations are very inconsistent. The one closest to my house is jacked up. Won't show wind speed and the wind chill is alway 0 degrees. Which on a 60 degree day seems a little off.
My personal favourite is Accuweather for iPhone and Accuweather for iPad...both were free and have a full list of features including animated radar images and video forecasts. They are easy to navigate and have very accurate information.
Not using an app, but I do use weather.com to send texts for sever weather alerts. Aside from that, the built-in widget does the trick for me. I don't generally care about the in depth details. Just need to see if there is sunshine, rain, snow, etc. on the slate for the day//week.
I use WFMY. Its out of Greensboro,NC but you can set your location. Really cool. Storm tracker to show you the direction of each cell. Hurricane and tropical storm tracker and Earthquakes. We have a lot in the US!!