SQL Server Migration using a DB Mirror (Rolling upgrade)
We used successfully mirroring to migrate databases from SQL 2005 to SQL 2016 (on new hardware). All worked flawlessly. On hardware that was 13-14 years old and using Windows Server 2003.

Can't say anything about MS support, as since we had such an old version, we didn't have any type of support from MS (not even Extended).

If you're curious, here's a short list of steps we used:

Prerequisites


logins
linked servers
jobs
SSIS packages
export sys.messages


Mirroring preparation


setup endpoints and permissions
disable log backups
restore dbs on mirror
mirror setup on both the principal and partner
re-enable backups


Actual migration - the only moment of downtime


manual fail-over, make sure you do it step by step, as it will take a while to get in sync
break mirror and set dbs online
set the databases on the old principal as OFFLINE, so the apps won't connect to it
UPDATE STATS (fully if possible) - can't stress how important this step is
then enable the application logins and you can get on with it


Everything gets a tad more complicated with replications, but you get the point. It works very well. Log shipping will work too, no worries. But since it involves a bit more manual steps, we preferred to reduce the complexity if possible.