I have a setup running now where I use the two GPIOs of an ESP-01 to control one of those 5V double relay modules. The box is powered from an external 230VAC-12VDC supply. Then in my board with the ESP I have a small DC/DC buck step down converter that provides 5V for the relay module. Then (I'm lazy) a miniature module with a 1117 provides 3.3V for the ESP.
I have not seen the need even for decoupling C's, as the unit runs nicely without.
It has not been running for long, but I haven't seen a reset yet.
My theory is that those little buck converter boards do a great job in providing the current needed with a fluctuating load. Unfortunately I do not have a scope, so I cannot investigate further, I just see that it works.
And they are cheap; you get twenty for 10$:
http://www.aliexpress.com/item/20-pcs-Ultra-Small-Size-DC-DC-Step-Down-Power-Supply-Module-3A-Adjustable-Step-Down/32262311443.html