As far as I know, desirable is practically always compression since the surface - coated layer or generated zone of the same material - is produced to protect the surface, i.e. it should not form cracks which causes corrosion or other failures. If possible it should repair itself, cf. our teeth. This means that the layer below is under tension since everything has to be in equilibrium. For a multi-layer system this might have a complex stress-strain construct especially since some layers are less strength than other. Too high stress can also damage the entire system (spalling observed in coated materials but also in quenched and tempered surfaces like railroad tracks) so that the investigation of stress is essential in practically all technical fields. 

Dear colleague,

Materials properties are normally  tuned,  targeting  a specific application. Lower or higher stress/tensile strength, everything has a particular application.

These properties can be modulated following "Structure zone diagram". For more detail about this, kindly check this literature

A structure zone diagram including plasma-based deposition and ion etching

http://www.sciencedirect.com/science/article/pii/S0040609009018288

this will certainly clear your doubt.

When you say internal stress, is there much difference between external and internal stresses in a thin film? Internal as in in-plane or through thickness?