The questions first: What is the proof of resolution of singularities that you know?

Why am I asking?: There are a number of proofs of resolution of singularities of varieties over a field of characteristic zero, all with more or less similar flavor but different in technical details and in choices that the resolution algorithm allows us to make. When writing a proof that uses specific features of some of those details I can't stop being uneasy about assuming the reader read about the specific constructions elsewhere. I would like to know from MOers what proof you have seen and if you have a reason for the choice, if it was a choice, I would like to hear it too.

Maybe asking about what you know is too invasive. I am just asking for the proof that you happened to find in your way, even you have only read a few lines of it.

The purpose of the question: The conspicuous one. To get a sense, by a rough approximation and a small sample, of what proofs are more culturally known. Have a concrete feeling when sending a reader to find the details in other paper, either of feeling OK with it or of guilt.

It is a question about fashion, which also has its role in mathematics... and knowing what the fashion is is useful.

What details?: Although I had in mind a specific detail of the proofs I didn't mention it because it is not the only one that changes from proof to proof and because the result of the poll gives information about all of them. Examples are: the resolution invariant, the ways of making the local descriptions of the centers of blowings-up match to form a globally defined smooth subvariety, the ways of getting functoriality and the different meanings that functoriality can have...

It is a community wiki question, so feel free to change what is said here if needed or if you want the poll to also give information about other questions that you would like to be answered. (or for correcting the English!)

The questions first: What is the proof of resolution of singularities that you know?

Why am I asking?: There are a number of proofs of resolution of singularities of varieties over a field of characteristic zero, all with more or less similar flavor but different in technical details and in choices that the resolution algorithm allows us to make. When writing a proof that uses specific features of some of those details I can't stop being uneasy about assuming the reader read about the specific constructions elsewhere. I would like to know from MOers what proof you have seen and if you have a reason for the choice, if it was a choice, I would like to hear it too.

Maybe asking about what you know is too invasive. I am just asking for the proof that you happened to find in your way, even you have only read a few lines of it.

The purpose of the question: The conspicuous one. To get a sense, by a rough approximation and a small sample, of what proofs are more culturally known. Have a concrete feeling when sending a reader to find the details in other paper, either of feeling OK with it or of guilt.

It is a question about fashion, which also has its role in mathematics... and knowing what the fashion is is useful.

What details?: Although I had in mind a specific detail of the proofs I didn't mention it because it is not the only one that changes from proof to proof and because the result of the poll gives information about all of them. Examples are: the resolution invariant, the ways of making the local descriptions of the centers of blowings-up match to form a globally defined smooth subvariety, the ways of getting functoriality and the different meanings that functoriality can have...

(edit) Forgot the "request for advise": If you have would like to give advise about how you have dealt with similar situations and describe your example that is welcomed.

It is a community wiki question, so feel free to change what is said here if needed or if you want the poll to also give information about other questions that you would like to be answered. (or for correcting the English!)