I'm looking for an Android application that can mute the sound notifications of all applications except one.
E.g. on Windows 7 I can do this in two clicks:
I use a Samsung Galaxy S3 with Android 4.1.2. Free is better.
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It seems like DetoxDroid could fit that bill. It's intended for "digital detoxing", and one of its features is a global DND mode ("mute all applications"). It lets you define exeptions ("…except one"), thus matching your requirements.
For now, it's not available at any store I know of, and APKs must be downloaded from Github and side-loaded. In a few days it should become available on F-Droid, though (I've just processed its merge request today). To get the required powers, you must grant it once the
WRITE_SECURE_SETTINGS permission via ADB – then it works fine.