UPDATE: 2018-08-18
Raspberry Pi 3B+ supports PoE through the use of a HAT.
Raspberry Pi Zero with PoE? It looks like there aren’t any ready commercial solutions, you have to build one yourself. Ah, just like the early PCs, if you wanted to connect to an Apple computer or printer over serial.
20180620/DuckDuckGo poe raspberry pi zero
20180620/https://lifehacker.com/[object%20Object]
20180620/https://n-o-d-e.net/ethernet.html