I have installed the Ubuntu Mate Image for the Raspberry Pi that I found here https://ubuntu-mate.org/raspberry-pi/ - During the boot process the only thing that appears on screen is the rainbow screen and a yellow lighting bolt on the right top.
The Raspberry Pi 3B+ has new hardware for Ethernet, WiFi, Bluetooth and Power management.
As a consequence it requires new Firmware, different Device Tree settings and possibly kernel changes.
There is no current support for MATE on the Pi3B+. A number of people have managed to get the Pi3B+ working (or partially working) by installing the Foundation Firmware and kernel - but this is untested and the interfacing to software may have issues.
There is development on Ubuntu MATE for the Pi but it seems unlikely that there will ever be Ubuntu MATE 16.04 for the Pi3B+. Ubuntu MATE 18.04 will likely be using a standard Ubuntu kernel See https://ubuntu-mate.community/t/status-of-ubuntu-mate-18-04-for-raspberry-pi-3-b-b/18054/6
Do you REALLY need MATE? MATE is not officially supported by Canonical and the Pi support is marginal at best.