Guiding Principle: Focus on the audience
So many resume writers forget that there is a “customer” on the other side of a resume submission. What you send depends on how you can help the “purchaser” make a good decision – e.g. talk to you – and make it as easy as possible.
With an Applicant Tracking System (ATS), it is always a guessing game. There are a few that accept multiple documents and the sender can manage their portfolio of submitted documents (e.g. multiple forms of a resume, cover letter, bio, etc.). However, in the general case, a cleaned up .txt version will not only go through the ATS with ease but, if properly formatted, provides a document that a recruiter/hiring manager can read and follow easily. One document is plenty, and this is the one.
The best way to get to a reformatted .txt is to create an intermediate unformatted resume in MS Word that doesn’t get sent to anyone. I have named this document UF, for unformatted. It is a working document to make reformatting of the .txt version easier. The UF document will go through an ATS in most cases but since it only takes a second to create a .txt, why not just make the best choice.
Regarding multiple copies, this is primarily in my opinion for sending attachments via email, especially to recruiters. The 1st attachment should be the PDF in case the recruiter is opening documents on a mobile device (phone, tablet) or on technology that is not compatible with yours (older version, Mac to PC). MS Word documents do not display as well there. People tend to open the 1st attachment 1st, so the PDF goes 1st.
The next document should be the MS Word version, the “workhorse” resume. Recruiters need that one too. Don’t attach it the UF version. The good formatted MS Word version and the unformatted MS Word working version will have the same icon on the attachment line and will be confusing to the recipient.
Finally, attach the .txt. If the recruiter needs the .txt version for their database, they will recognize it and know what to do. Otherwise they will ignore it. Keep in mind this is a customer service strategy – making life easy for the recipient.
If the resume is going to a networking contact, friend, etc. the PDF is usually a good choice.