TITLE: What formal properties should resolution of singularities have?
QUESTION [15 upvotes]: If I were going to propose a new construction as a "replacement for resolution of singularities", what properties would my replacement have to have?  [I am going to do no such thing -- this is purely speculative.] Is there a shortish list of theorems such that any construction verifying the properties on the list would thereby deserve to be called a resolution of singularities?

REPLY [6 votes]: It should exist in all characteristics, even mixed characteristic! 

This sounds somewhat cheeky, but I was fairly serious. To algebraists, and the OP is one last time I met him, a purely ring-theoretic statement which can largely only be verified in char. $0$ feels more incomplete than to a geometer. 
This is a main reason why the use of resolution of singularity is restricted in attacking some of the open questions in commutative algebra: often the hardest case is mixed characteristic, and if you are extremely lucky and smart you can reduce it to a statement in char. $p>0$, then you are dead!
Because of the above reason, I would mention that de Jong's alteration  has found some spectacular success in commutative algebra (so one can  replace birational by surjective and generically finite). A main example is Gabber's proof of the non-negativity part of Serre's conjectures on intersection multiplicities (see number 1 here for an expository account) .