I don't know Stockholm well enough to get into its suburbs yet.
For Copenhagen, where I lived until recently, I always liked Lyngby; had I stayed, I would have considered living there.
It has a charming main drag area, a major transit hub, and some beautiful streets.
Main street:
https://goo.gl/maps/L4svT6j9eNcEdim88
Lyngby Church (12th century):
https://goo.gl/maps/aMnRHDD3itQCh1MHA
Quiet streets:
https://goo.gl/maps/cqjcPZ1tmC5Mgw7y8
https://goo.gl/maps/ukRTYwuxn4kKWMiU8
https://goo.gl/maps/aUUxrudCNrhdedEE8