Are there any places where I can find a publicly-available electric socket in Munich? I'm trying to charge my dead mobile device.
Short answer: libraries
Here you can find the list of all libraries in Munich and its surroundings.
To elaborate more on chirlu and Willeke's comments, every district in Munich has a library/libraries, every library has a lot of free for use sockets and you don't need a subscription or user card to use the library space.
Here you can see the list of Münchner Stadtbibliothek branches. There are about 30 branches in Munich. Most of them close at 7pm, have shorter opening times on Saturdays and are closed on Sundays. The map of the branches is provided below:
Here you can see the list of 14 LMU libraries.
Here you can see the list of 9 TUM libraries. Half of them are actually in Garching, Freising and Straubing.
And if one lists the libraries in Munich, the Bavarian State Library should be included by default.
By now most shopping malls offer lockers that have charging cables. They are tiny locker thingies where you can pay a small fee or just leave a coin deposit and put your phone inside to charge it. You'll get a key and can pick it up when you're done shopping. I did a little research on the mayor shopping malls that the Munich city council lists on their website.
Riem Arcarden says they have power sockets throughout the center. Scroll down, click on Anderen, then on Steckdosen.
Sie wollen Ihr Mobiltelefon laden oder mit dem Laptop arbeiten? Wir haben für Sie überall in den Riem Arcaden Steckdosen. Hier können Sie und Ihr Gerät "auftanken".
This roughly translates to:
You want to charge your mobile phone or work with your laptop? We've got power sockets for you all over the Riem Arcaden. You and your device can "refuel" there.
Olympia Einkaufszentrum does not seem to have lockers or power sockets. You can ask at the information desk there.
Einkaufscenter Neuperlach also does not seem to have those. Again, you can always ask.
Fünf Höfe doesn't list them either.
To give you an impression what those phone charging lockers look like, you can check out the one in Alexa in Berlin here.
Cafes like Starbucks and Balzac will have lots of power sockets all around the place. They target young people and (this is a bit biased, sorry) hipsters with Mac Books and they don't mind you using their power sockets. Often those places are so full and the staff so little engaged that you can simply sit down upstairs without even buying something, or just getting the house coffee or something like that.
Public transport does not usually have any chargers available. The Apple Store might. They sure have free wifi, leading to hordes of kids and the occasional refugee sitting in the vicinity of the Apple Store enjoying the wifi.
In Restaurants you should definitely ask for permission, but if you're a customer that is hardly ever a problem.
Then there are also sometimes Deutsche Telekom stores (the magenta ones) that rent out charged power banks, sometimes for only a deposit. I've seen that in at least two cities, but not specifically in Munich yet.
If there are USB sockets anywhere, always keep in mind that using an unknown source with your phone poses a security risk. For details of what I'm talking about, see:
I guess you want power for small consumer devices, not cars, and nothing where you need special equipment to get an acceptable voltage etc.
Then you're probably out of luck. While there may be exceptions (I can't check 100% of Munich, of course), public and free power sockets are not something known or expected to exist. While there are plenty that you could reach (without entering private homes etc.), none of them is supposed to be used by strangers. It's just a good chance to get someone angry with you if noticed.
Feasible non-free alternatives are cafes/restaurants. Asking if you can use one near your table (if there is one where you're sitting), while drinking/eating something, won't be denied.