The filter capacitors in the power supply are connected to ground, so when the ground pin isn't connected, they apply voltage to the chassis and the shields on the input connectors (possibly quite a lot, up to half the mains voltage). Because the oscilloscope reads (Vtip - Vshield), the relatively large voltage now present on the shield becomes a term in that expression, and unless Vtip follows it closely, you're going to see noise.
It's not really unsafe to ground it like that for signal purposes only IMHO (others may disagree on this topic, we've had lots of arguments about it...), but you must not trust it for safety, because you don't know how good of a ground it is.
You really need to get proper grounding installed for your lab. As I'm sure many people will point out, grounding isn't absolutely essential for safety, but grounding items that are designed to be grounded is.
Keep in mind that when the chassis isn't grounded, that voltage comes through the PSU filter capacitors, so the only thing separating you from the equivalent of sticking your finger in a light socket is a little capacitor. Do you trust it? Granted, they're probably good ones in that brand-new scope, but that doesn't necessarily mean they're safe.