Oddly, I got similar behaviour from my Electron, and the Spectrum I have can do this at power up. With the Electron, it was a frayed/damaged cable just where it joined the plug. If the cable wasn’t quite at the right angle, all I got was some random blocks of junk on the screen (clearly not enough power getting to all the bits inside). Cutting off the old plug and putting on a new one solved that problem.
The Spectrum seems to have a loose grip on the power plug. This is inside the socket, so replacing the plug won’t solve this. If the plug gets wobbled or it’s not seated right at power up, I get random characters too. So maybe your Spectrum has a similar problem.