Examples of calculating perverse sheaves on algebraic varieties with easy stratification

I would suggest you two papers

Gudiel Rodríguez, Félix; Narváez Macarro, Luis: Explicit models for perverse sheaves. Rev. Mat. Iberoamericana 19 (2003), no. 2, 425–454.

This deals with the two-strata case. And

Gudiel-Rodríguez, F.; Narváez-Macarro, L.: Explicit models for perverse sheaves. II. Algebr. Represent. Theory 11 (2008), no. 2, 149–178.

This treats the case of an arbitrary stratification. The techniques are more complicated but it extends the basic ideas of the previous paper.

Geordie Williamson has a very nice set of notes on perverse sheaves: http://people.mpim-bonn.mpg.de/geordie/perverse_course/lectures.pdf it deals with some examples on curves (section 10).

You can also look at De Cataldo and Migliorini's paper (section 2.2): http://www.ams.org/journals/bull/2009-46-04/S0273-0979-09-01260-9/S0273-0979-09-01260-9.pdf

Things are easiest when the automorphism group of $M$ (with its stratification) acts with finitely many orbits on $T^* M$: see Perverse sheaves on Grassmannians, by Tom Braden.