This is how the voltage drops on my ESP-01 module almost 300mV. It's repeated every 100ms.
This is with the module connected to my wifi, but doing nothing.
I'm literally powering the ESP-01 directly on the connector with an ATX PSU, so power shouldn't be an issue.
Putting a cap on the supply won't do much if it ain't like 10mF or so.
Is this a ESP8266ex issue or is it a ESP-01 (from AI-thinker) issue? Or some thing else?
Do you experience the same with other modules?