How stable is the "customer" power on a commercial aircraft?
There is a pretty great article on the design of the 777 electrical system here.


  Power for avionics needs to be clean to ensure there are no spikes or
  dips that would cause avionic failures as this could lead to otherwise
  avoidable emergency declarations.


The avionics and passenger power systems are two completely different systems. The 28V instrument power system is pulled right from the source, 


  Flight instruments on the 777 are primarily powered by 28 Volts DC.
  Direct current power is supplied by three-phase transformer-rectifier
  units. There are four such units each rated at 120 Amperes.


This pulls right from the generator/alternator output of the plane. Everything ranging from your classic Cessna 152 all the way up through a 777 regulates their alternators/generators with voltage regulators and fuses to protect against current overloads. These things work in tandem to protect the avionics and internal power systems of the plane. The 28VDC system for the avionics is also connected to the battery system which prevents dips should the generators have issues outputting enough power or quit all together.   

The rest of the plane is powered by 110/200V 400Hz system. 


  The primary electrical system is a 3-phase, 4-wire, constant frequency
  400 Hz, 115/200 Volt system, which has been the industry standard on
  commercial jet transports since the Boeing 707


Interestingly it's 400Hz, I'm still looking to see how they get that to 60Hz.




  Since the plane is already providing clean power to the avionics, the
  power coming out of the transformers to step it down to 110V, 60Hz for
  the outlets should also be pretty clean.


The power is quite "clean" but you have some incorrect assumptions here. The power is not really stepped down. That is something that occurs on the power grid since most power lines on the street carry a significantly higher voltage than what you have in your house plugs. In this case we have the luxury of choosing an alternator/generator system that is close to (most likely slightly above) our required voltage. Slide 4 from this presentation from Boeing alludes to the fact that the system outputs 115V (and is presumably conditioned) and is then split to the cabin as well as the 28V DC bus. Power does not come out of the transformers it comes out of the generator/alternators it then is conditioned and transformed as needed. Stability is achieved though numerous conditioners and regulators, see the article I linked for more details on them.  

Don't worry, it's safe to charge your laptop on a jet.