One of the good instruments that I know is the Shack–Hartmann wavefront sensor:

http://en.wikipedia.org/wiki/Shack%E2%80%93Hartmann_wavefront_sensor

Analysis of data obtained by this sensor can be very usefull to understand abberations in the system. Which additional methods can you recommend?

Shack Hartmann is good and fast but has limited resolution depending from the total number of lenslets. I think the better you can achieve is lambda/20 resolution usually. A variety of wavefront sensors are available. Personally I used Twymann-Green interferometer to characterize my adaptive optics system before switching on the closed loop with shack hartmann. Fizeau interferometers are the standard for high quality wavefront measurements which can go to lambda/200. The resolution is a factor 100 or more in comparison to shack hartmann but are inerhently slower and more expensive.

Depends upon your system and application.  I suggest getting a copy of "Optical Shop Testing" edited by Daniel Malacara.  The 3rd edition is the latest, but any edition will do.  You will see more ways to test aberrations than you thought possible, starting with a simple knife edge and working up in complexity to interferometers. 

The Shack-Hartmann wavefront sensor is a good solution.  There is a tradeoff between spatial resolution (dependent on the lenslet diameter and number of lenslets) ,dynamic range and sensitivity (measurement accuracy).  A shorter focal length microlens will yield a larger dynamic range, but with reduced sensitivity.  Some standard products have lambda/250 sensitivity with a dynamic range of 115lambda.

I agree. Depends a lot on application and accuracy. If you are interested in "normal" aberrations (third order Seidel) you will find plenty of methods in Optical Shop Testing, from nm to m accuracy. If you resconstruct the wavefront, do not be obscured by the value of "20th Zernike polynomial", or "tetragonal astigmatism". Only a few aberrations have clearly defined physical senses, and enable to introduce reasonable corrections. The rest is mathematical fitting of an infinite series.