Forgive me if this sounds dumb.. but have you tried moving your scope probes and leads and even meter away from the psu ? probes can pickup RF from the circuit causing voltages which are not really there.. one angle on the probes may pickup RF, another angle may not.. also some RF can even cause the meter display itself to register voltages which are not really there.. also make sure you are not touching any part of the circuit / probs (can touch gnd but not the probes themselves) Just a thought...
EDIT:
Its also possible the small track length on the pcb could be picking up RF which may explain why you see voltage there.. it may not even be 0.6V, its just RF noise which your meter is seeing and translating to 0.6V on the display.. I don't know how long the track is.. but nothing surprises me much anymore..
EDIT2:
Which side of the zener did you disconnect, the transistor side or other side ? The zener could act as a antenna and float the Base voltage up, may be then internally clamped to 0.6V in that case... again just a thought..