ESP8266 Supply Voltage dropping below 3.3V
The ESP8266 will do an analog calibration when it is running the user code. This will typically be the worst case power requirement. The peak current requirement can be as much as 1/2 an amp. While your regulator probably can supply that I would bet your interconnecting wiring, and maybe the power source, are causing too much voltage drop. 

The fact that you can see the voltage go down to 2.9 volts makes me thin that you have serious problems.

Early on when I started with the ESP8266 I was getting flaky operation. I put it down to bad code. But eventually I ruled that out. I took my setup to work (where I design electronic products) and looked at the power connections with a digital scope at the module. Right at the last point before the traces run under the shield. (I use the ESP-12 modules) It dipped just under 3.0 volts for two microseconds, and that was enough to cause a reset. 

These things are not tolerant to any power dips. And this is why I recommend anyone starting with the ESP8266 to buy a Nodemcu or a Wemos type module. Something that has the regulator right at the processor.