What could be causing this sine on 7810 output when powering BLDC fan?
This all looks perfectly normal and should be expected.  Look up the load transient response spec for the regulator.  As the motor is commutating, the load it presents to a constant voltage changes suddenly.  The regulator can't change its output current quite that suddenly, so for a short time the voltage is out of regulation.  Then the regulator catches up and the output voltage goes back to being nicely regulated.  Again, this is all to be expected.

If you are just driving a motor, there is no problem here except perhaps for using a linear regulator to supply the motor voltage in the first place.  That is going to waste a lot of power as heat in the regulator.  Wasting the power may not be a big deal, but dealing with the heat often is.

The only problem is if you are trying to use this 10 V for something else where that ripple matters.  The first way to attack that is to use a larger output cap on the regulator.  100 nF is very small, especially for something with high current transients.  Nowadays there is no point using something that small.  My minimum would be 1 &micro;F unless the regulator would not be stable with that (in which case using a different regulator is probably a good solution).  Just because this regulator is specified to be stable with 100 nF minimum, it doesn't mean that's a good idea.

A better solution all around might be to drive the motor from the higher input voltage and use PWM to make the apparent average motor voltage what you want.  Beyond a few 100 Hz, the motor won't know the difference.  Some motors make audible whine at the PWM frequency, so often 25-30 kHz is used since it is just above the human hearing range.