IMPORTANT NOTE: For Android 4.1 Jellybean (API Level 16) and above the READ_LOGS permission must be granted in order to display all logs. Therefore, the app will run a wizard to make it easier: You can make it with root or with ADB. For ADB the following commands are necessary:

"adb shell"
"pm grant com.anrapps.ultimatelogcat android.permission.READ_LOGS"

This app is 100% Open Source, check out: https://github.com/bamsbamx/UltimateLogcat

Ultimate Logcat lets you find and fix almost all problems with your phone. It displays the Logcat output and lets filtering the logs according to their log level, system, main or radio events.

Also, it contains Crash Finder. A service which will run looking for any problems in any app and displays a notification when a crash occurred. It also lets you to save the complete stack trace

The following are the filter possibilities:

LEVEL:
-Verbose
-Debug
-Information
-Warning
-Error
-Assert

FORMAT
-Brief
-Process
-Tag
-Thread
-Threadtime
-Time
-Long
-Raw

BUFFER
-Main
-System
-Radio
-Events