Power Issues with an Arduino Uno and ESP8266

I am trying to run off of an external power supply. The three I have are as follows:

One that supplies 5 volts with 500 mA current

One that supplies 9 volts with 500 mA current

And finally, one that supplies 14 volts with 400 mA current

According to Arduino's website the only one that should not work at all should be the 5 volt one. However, that is the only one that works satisfactorily. If I use either of the ones that should be more suitable, the ESP8266 will not boot, but the arduino will. Any ideas why? I've already cut up the 14v one, and I don't want to have to take apart the 5v one to run it into the project's housing. I measured the voltage to the ESP, and it's exactly the same between every power supply I try.

What ESP8266. There are many.
How did you power the ESP8266. From the 3.3volt pin of the Uno?

The 14volt supply is dangerous for the Arduino. Do NOT use.

Where did you connect the 5volt supply to. Only the USB socket and the 5volt pin are valid options.
The 9volt supply should be connected to the DC socket.
Leo..