What properties of a canonical bundle are preserved under birational isomorphism between
smooth projective varieties. In particular, is triviality of the canonical bundle preserved?



Birational properties of canonical bundle 
Generally speaking, to your specific question, the answer is no. 


See for example Hartshorne, Chapter V, Section 3. On the other hand, numerous other properties are preserved (some of these are also mentioned in Hartshorne, for example, the geometric genus). There are more subtle things as well, see any book on the *minimal model program* for example. 


With regards to your original question, you might want to try looking up *crepant resolutions*.


