Here are some pictures.
In the evening I was thinking that near home (300~500m) there is a FM transmitting antenna on 105MHz and 93.3MHz. It's very powerful: with some FM radios I can't receive some other stations, because signals take input stages into saturation.
However, the track is noisy also with the input coupling selector set to GND: if the problem is due to RF interference, it is a shielding problem near the AD converter, because RF can't enter from the BNC.
I would agree that most likely something gets into the signal path well behind the inputs, near the A/D converter. The fact that you can get the same amplitude of the noise/interference signal at either 5mV/div or 1V/div is consistent with that assumption. It probably also makes no difference at all whether you switch the 20 MHz bandwidth limit on or off? The 125 MHz frequency you see seems not directly related to the FM frequency. It might be a subharmonic of the 1 GHz or 500 MHz sampling frequency?
My own DS1054Z certainly does not show this kind of behaviour. Testing yours in a location away from the FM transmitter sound worthwhile -- you probably don't have to travel 300 km, but a few km might already make a difference? If that reduces or removes the noise, I would agree with your conclusion that proper shielding of a scope should take care of such interference. But will another DS1054Z fare better in that respect? Having Radio Vatican in your backyard is an untypical challenge, which Rigol may not have considered when designing the scope ;-)
I think it is worthwhile to check the firmware version. My scope did not come with the latest firmware when I received it a month ago. When you go to Utility - System - System Info, it should display the software version 00.04.02.SP4. Rigol did fix a jitter issue in that latest firmware, by changing the PLL operating parameters. While I don't have a theory how that would impact your noise issue, it is certainly an improvement worth trying. The firmware is officially available here:
http://beyondmeasure.rigoltech.com/acton/fs/blocks/showLandingPage/a/1579/p/p-0019/t/page/fm/0, so installing it should not affect your warranty.
By the way -- the built-in screenshot function of the DS1000Z series is really nice and simple. Just plug a FAT32-formatted USB stick into the front USB connector, and each time you press the green "print screen" button, the scope will store a screenshot on the stick.