I purposely make my own application crash. I get the following dialog.
Use the acra library in your app. This basically "is a library enabling Android Application to automatically post their crash reports to a GoogleDoc form". You can modify its use to pop up a dialog like the above, and ask the user to report the issue, with an additional user-typed message, even.
As far as I know, the Report
button only shows when the app is installed via the Market (i.e. is signed and not in debug mode nor with a debug key). This data is all shown in LogCat if you are USB debugging.
Edit: Also, you can catch this information and file a report yourself if you want, using this method.
Your phone may running a customized rom like CM. If you want to report log when your app get FC,one way is catch exceptions and mail them.