How do I verify that my 3.3v power rail meets the requirements for an Ember EM357 SoC?
It's more important to know how much current the module can draw. If your DC source isn't up to the task, the module will collapse the rail and all sorts of interesting behaviour can ensue (hiccupping, brown-out, etc.)

Industry practices for power supply ripple measurements must be done with a short probe (no long ground lead) or a direct-connect coax with proper termination, with 20 MHz bandwidth limiting turned on, and a combination of ceramic and electrolytic capacitors at the measurement point to attenuate CM noise and give you a 'true' indication of what the ripple is. In industry, it's common to use a 10 microfarad electrolytic capacitor in parallel with 100nF ceramic. 

How much ripple your module can actually tolerate is hard to judge unless the manufacturer has specified it. Less is generally better when it comes to microcontroller-based gadgets, especially ones that do a lot of analog-to-digital conversion.