When do you use 'nom de plume' vs. 'pen name' vs. 'pseudonym'?

Dictionaries usually treat nom de plume as synonymous with 'pen name' or 'pseudonym'. Example from Merriam Webster's dictionary:

Definition of nom de plume: a name that a writer uses instead of his or her real name : PSEUDONYM, PEN NAME

In what context would you use one rather than the other? Is it a matter of regional preferences (ex. one is more prevalent in Britain and the other in America)?

Solution 1:

A pseudonym is any time you are using a name other than your own.

A pen name is when an author publishes under a different name.

A nom de plume is when a pretentious author publishes under a different name.

Generally a pseudonym has the intention of concealing who the real person is. With a pen name or nom de plume it may be for purposes other than concealment.

Solution 2:

I would suggest you write under a pen name rather than a nom de plume. The French don't say nom de plume: they say Nom de guerre. Nom de plume was made up by English-speakers 150 years ago. As H.W.Fowler said (and Etymonline mentions) it is "ridiculous for English writers to use a French phrase that does not come from France."

As Stuart F said, pseudonym is more general. You can write, or do anything you like, under a pseudonym.