I can't speak to the question of which singing squares are popular
in any particular community (and I rarely attempt them myself,
since I'm not much of a singer), but for what it's worth here are
a few resources for callers looking for singing squares in the
traditional (as distinguished from MWSD) style.

Rich has already said "I am familiar with the ones on Ralph Sweet's
CD and Bob Dalsemer's two collections," meaning Ralph's _Shindig
in the Barn_ CD and Bob's _When the Work's All Done_ and _Smoke on
the Water_ albums.

Ralph's book _On the Beat with Ralph Sweet_

https://store.cdss.org/books/product/1052-on-the-beat-with-ralph-sweet?search=on%20the%20beat%20

includes over thirty singing squares with printed music, calls,
and, in some cases, teaching tips and/or notes about the music.
The book also includes a nice collection of patter squares.

Nils Fredland, who assisted Ralph Sweet with his "On the Beat ..."
book, has also edited a book titled _New River Train_

https://store.cdss.org/product/1046-new-river-train

with music, calls, and notes for forty singing squares from the
repertoire of the late Keith Blackmon of western New York.

The CDSS websites Square Dance Resources page

https://www.cdss.org/resources/how-to/square-adance-resources

includes a "SQUARE DANCE VIDEO AND AUDIO" category with, among
other things, 25 videos of singing squares and some links to
other resources.

This page on the Square Dance History site

https://squaredancehistory.org/items/show/881

will lead you to seven videos of singing squares called by Debbie
Gray and the Deb-U-Tones (back-up vocalists Alice La Pierre and
Julia Huestis) at a dance in Cambridge, MA, in 1989. The videos
(at least the sampling I've just looked at) include walk-throughs.
[Two points to note: (1) Debbie's walk-throughs seem quite well
suited for the dancers she was calling to. Beware that contra
dancers in many communities in 2017 are less familiar with
squares than Boston dancer were in 1989. (2) The vocalists in
the video are sticking together with accurate timing. Group
vocals by people who are less well rehearsed can easily become
unintelligible. Also, I think Debbie's mike is s little hotter
than the others.]

The Square Dance History site has lots of other pages with
the tag "singing square"

https://squaredancehistory.org/items/browse?tag=singing+square

Some of these have clips of modern western squares, but
there are also quite a few with material suitable for
traditional-style dancers.

If you can find a copy of the album _Square Dance Party with
Ed Gilmore_, it includes several singing calls suitable for
traditional-style dancers. I have this as a 33 1/3 r.p.m.
12" LP album packaged with printed calls and explanations.
I believe it was previously issued as a set of 7" 45 r.p.m.
discs. I don't know whether the material has ever been
reissued in a (non-bootleg) CD or mp3 version.

Walter Lenk has a collection of singing squares here:

http://www.configular.com/SingingSquares/
( also linked from https://squaredancehistory.org/items/show/812 )

In includes about forty dances, grouped into four categories
ranging from Walter's favorites to ones he pretty much never
calls, plus a link to even more in a collection that Walter
transcribed from the call book of the late Chip Hendrickson:

http://www.configular.com/SingingSquares/Chip_Hendricksons_CallBook.pdf

Finally, a note of caution sbout learning singing square from
printed sources: I don't know about the rest of you folks,
but when I read the calls for a singing square without having
heard them either live of on a recording), I often find myself
slightly puzzled about how some of the syllables are supposed
to be matched up in time with the notes of the tune. In some
cases, I think it would be quite easy for a reader to come up
with an interpretation that doesn't quite match what the author
intended, even if it's not as consistently off as the mismatch
of lyrics and melody demonstrated here:

https://www.youtube.com/watch?v=RYjIj23-7D4

--Jim