Is it ever bad to run an IC VCC trace next to a ground plane with 1mm clearance?
In general, it's good to run Vcc as close as possible to the ground copper that will carry its return current. This reduces the size of the current return loop, which minimizes radiated emissions and improves radiated susceptibility.

If the potential on VCC might be over 50 V, then you need to start thinking about creepage and clearance distances. But I doubt this is the case here, since you're talking about powering a microcontroller.