No VFD glow can mean the seal is broken and the glass has lost vacuum. If that is the case then it is done. Inside the display in one corner you should see a black patch/splotch approximately 1/4 inch or 5 mm wide. This is the "getter", a reactive metal intended to remove (react with) any small amount of oxygen left inside after creating the vacuum. If this patch has turned white-ish it is a clear sign that the seal is lost and the VFD is dead. Post a photo if you are unsure.
Also, the waveforms on the filament don't look right. You should see a clear square(ish) wave of a few volts P-P. The 7 kHz frequency sounds ok, it's a reasonable frequency for this DC to AC converter. The short spikes you are seeing could be what you see at each edge of the square wave, if the ground lead isn't connected. Then you would only get the very high frequencies (in the fast rise edges) through capacitive coupling only.
If you power the VFD from a floating power supply then you can hook the ground lead to one end of the filament and the probe tip to the other. This is a little risky since I'd the VFD power supply isn't truly floating you could short out the filament supply, potentially damaging the circuit. An alternate and safer probing technique is to use a dual channel scope, set to display channel 1 minus channel 2 (the difference). This provides a simple differential probe, which will work just fine at the frequencies involved here (don't expect much at multi MHz though). The vertical V/div should be set the same for both channels. You can test this psuedo-differential probe using the scope Cal output. Connect one probe tip to ground and the other to the cal signal, and you should see the expected square wave. Then connect both to the cal signal. You should ideally see nothing, but you will likely see just a little ringing/glitching at the fast rise edges of the square wave, due to the imperfect common-mode rejection. You may tweak the compensation trimmer of one probe to minimise the visible waveform. Now, Connect both probes' ground leads the circuit ground, and probe both ends of the filament. You should see a square-ish wave.