What is the correct and/or most used order of the words in the phrase
comprised of the three words "coulda", "shoulda", and "woulda"?

Interesting question! I've heard this phrase in several permutations.
There is no single "correct" form of a colloquial phrase like this
one; I suspect that many language experts would probably scream and
run away if asked to endorse "coulda" instead of "could have" or
"could've."

But, even in the absence of authoritative sources, there's a way to
get a look at frequency of usage. Since this is a common slang phrase,
it's all over the Internet. A good way to check on the popularity of a
phrase and its variants is to note the number of results brought up in
a Google search.

And the winner is: woulda coulda shoulda.

Here's the tally from my searches:

32,400 results for "woulda coulda shoulda"

20,300 results for "coulda woulda shoulda"

19,900 results for "shoulda woulda coulda"

17,100 results for "shoulda coulda woulda"

 7,030 results for "coulda shoulda woulda"

 4,200 results for "woulda shoulda coulda"

Google Web Search: "woulda coulda shoulda"
://www.google.com/search?hl=en&q=%22woulda+coulda+shoulda%22

Google Web Search: "coulda woulda shoulda"
://www.google.com/search?hl=en&q=%22coulda+woulda+shoulda%22

Google Web Search: "shoulda woulda coulda"
://www.google.com/search?hl=en&q=%22shoulda+woulda+coulda%22

Google Web Search: "shoulda coulda woulda"
://www.google.com/search?hl=en&q=%22shoulda+coulda+woulda%22

Google Web Search: "coulda shoulda woulda"
://www.google.com/search?hl=en&q=%22coulda+shoulda+woulda%22

Google Web Search: "woulda shoulda coulda"
://www.google.com/search?hl=en&q=%22woulda+shoulda+coulda%22

I hope this is precisely the kind of information you're looking for.
If it is not, or if anything is unclear or incomplete, please request
clarification; I'll gladly offer further assistance before you rate my
answer.

Best regards,
pinkfreud

Thanks for the quick and accurate response, Pink.  However, some in my
office still aren't convinced.  I think they were hoping for some sort
of grammatically definitive answer (despite the grammatical shoddiness
of the words used).  But if it doesn't exist, what can you do?

Maybe I'm just weird, but I change the order depending on context.

If somebody says "I should have done xyz", I'll respond "shoulda woulda coulda".

However if somebody says "I could have done xyz", I'll respond "coulda
shoulda woulda".

In other words, I'll start with the word they used.  I find this rule
maximizes the annoying nature of this phrase -- which after all, is
the purpose.

Regarding all the "-oulda" variations, my own preference is "coulda,
woulda, shoulda." My logic is that it's shorthand for "If I could
have, I would have, and I should have." That order makes more sense to
me than the others. First comes the matter of whether or not the
action is possible. Then comes the choice to perform the action. Last
is the pronouncement that it was the right thing to have done.

It would be nice if there were someone like "Miss Manners" who could
issue edicts on the appropriateness of colloquial phrases. Hmmm...
maybe that's a niche I could fill. "Miss Pink, the Slanguage Person."

~Pink

I most commonly hear "coulda, woulda, shoulda."  Don't say it myself,
but this sequence seems euphonious enough.  There's no innate
grammatical order to the auxiliary verbs.  Trying to make a rule for
this is like trying to appeal to some law about the order of colors in
a striped pattern.

Archae0pteryx