On a few oscilloscopes, each channel will independently measure the voltage difference between the inside and outside of a BNC jacks without allowing any current to flow between the different jacks. More commonly, the outer ring of all the BNC connectors will be connected together. Typically, for scopes with a three-prong mains plug, they will be connected to earth ground as well.
From a safety perspective, connecting the scope ground and earth ground has a number of effects, some bad and some good. On the bad side, it means that when working with voltages which are substantially above earth ground, it creates a potential path for dangerous currents to flow which would not otherwise exist. If one were constructing a scope with a single input, this reason would probably represent a good justification for not connecting the input ground to earth ground. While it is true that having the scope lead be earth grounded could help avoid the possibility that an equipment chassis was live without someone knowing it, it's also entirely possible (if no other exposed objects are earth-grounded) that the even though the chassis was live it would pose no real hazard but for the scope ground itself.
The real safety advantage to earth-grounding scope leads arises when there are multiple probes whose ground clips are connected to different objects. Were it not for the earth grounding of the scope, connecting one probe's ground clip to something which was live (relative to ground) would cause the other scope's ground clip to also become electrically live. If that other clip was connected to an exposed metal surface which was not earthed, the entire surface could be dangerously electrified. Even if the other surface was earthed, touching the clip to it could cause sufficiently dramatic sparks as to startle the operator, thus resulting in the live lead touching other things it shouldn't. Nasty stuff. Having the scope ground clips be earthed would mean that the attempt to connect the scope lead to something that was live would quickly get noticed.
Conceptually, it would seem like the ideal thing from a safety standpoint (aside from having all probes isolated) would probably be a scope which had a two-poles-per-input resettable interruptor, and a safety circuit which would disconnect everything if excessive ground currents were detected along any path. Such a device could combine the safety advantages of preventing dangerous current flows between probes' ground clips, while simultaneously avoiding the introduction of a dangerous current path of its own. I don't think I've ever seen such a scope design, but it probably shouldn't be too hard to construct such a thing.