I agree that you'd probably see crashes but it still could be power supply related. Crappy SMPS designs do tend to shovel some noise from the primary into the secondary side as well as adding their own. And there's a lot of noise generally coming from an inverter as specified. It might just be adding noise to the line and crapping on the SnR occasionally. ADSL is quite robust but with noise it does slow down horribly if you bend it hard. I used to be able to slow my line down keying on my amateur radio transmitter back in the day.

About the best thing they did was adding a choke on it.

On top of that, it's a crappy 40MHz analogue scope. You're not going to see anything useful.