What causes random ringing in my inverter?
Random Ringing is due to "some" unknown external disturbance like a magnetic field from an old Weller soldering iron switching On/Off with a relay switch AC load current.  Lack of high impedance shielding on sense lines can contribute to this or poor layout.
( Notorious cause of glitches in the 70's/80's)


p.s. The well-damped bottom pulse after the disturbance suggests to me that your mains current sense ( make one) should see a spike here due to cross-conduction during the disturbance)

What is the rep. rate of this random disturbance?  1ppm ? only when low load?

Steady Ringing is an inherent response due to mismatched load to source impedances in a 2nd order or higher system response to a step input.  We can define this as Phase margin, Damping factor or Q and all are related to the equivalent circuit of the switched resistance before and during deadtime, the choke L, the wire inductance 1uH/m and the FET Miller Capacitance Cm * gm. 

For simple series RLC low pas filter I use  \$Q = \frac{1}{Rs} \sqrt{\frac{L}{C}}\$  and critical damping using this formula has Q=0.707 and has ~25% overshoot when Q=1

This is fundamental, but details are in the actual circuit model.

Wire inductance , crosstalk and stray positive feedback can amplify this Q factor as well by reducing the "phase margin". 

Possible solutions


Tight layout with a ground plane.   
Lead-lag filter on Vfb.   
Snubber on FETs for impulse during deadtime commutation.  
Complex realization of circuit impedances over every component, including all Caps that need low ESR L/R=T1, C*ESR=T2 for each component with conjugate matching., But too low ESR can reduce BW of feedback and reduce phase margin.


N.B.

I compute that your ringing is at 2.5MHz so if it is a x1 probe , it could be due to probe ground resonance but if 10:1 probe, that resonance is > 20MHz from long probe ground leads.

Speaking of disturbances

( We just had a 1-minute power failure in Richmond Hill near TO probably due to melting ice on underground HVAC cables and PD (partial Discharge arc) activity, which seems to happen when the snow melts annually in our area. )