Measure both sides of an AC and DC circuit
Yes, it's possible (likely, even). Most oscilloscopes have a common ground between channels, and common with earth. If you put two probes as you've shown above, the 12.6VAC in series with a diode will pass between the two probe grounds, which would be unfortunate (it could damage the probes or the oscilloscope). 

For power supply work, I use a Tektronix TPS2024 which has four isolated channels- isolated from each other, and isolated from earth.