I've tried several of these apps and there isn't one single app that does everything I want.
Sportstap - Good notifications, has the main American sports, but individual player stats don't seem to update until some time after the game is over. Has an ok widget.
Scoreboard - Has many more sports than Sportstap (AHL hockey is the specific one that I use that no other program has.) Notifications are better. It shows a baseball, basketball, hockey puck, etc. depending on what the notification is for instead of the single plain notification Sportstap has. But it doesn't have much of anything for game or player statistics and it updates pretty slow at times. It also has a better widget in my opinion.
Live Scores - Only has the main American sports, and doesn't do notifications. It does have great game and individual player stats updated throughout the game. Also, doesn't have a widget.
Sportstap is the most all around app but I'm a big fan of seeing individual player stats throughout the game. I use a combination of Scoreboard for notifications and Live Scores for game and player statistics.