Using flyback diode with relay, load regulation faults occur
There is nothing at all wrong with the drive circuit in your schematic, to the extent you have provided it, if anything it appears a bit overdesigned. 

I suspect you are seeing EMI through your relay contact to coil coupling causing the symptoms you describe. This would likely be related to other connections to your board and to stray inductances and capacitances, including your power supply which probably provides a low-Z AC path to the mains. Generally a tight layout using a multilayer board with full ground and power planes can help a lot. 

You cannot completely depend on the display of an oscilloscope in high EMI conditions especially if it is not a floating input type, so I do tend to discount your observations somewhat in that regard. 

If you have proper supply decoupling, resets coincident with relay contacts opening or closing are most likely caused by EMI coupling to input pins (including, of course, any reset input, but not limited to that particular pin).