That’s what I was going to mention. Here is Burn, Don’t Freeze which has two completely different sets of lyrics.
Taken to its logical conclusion in Joshua Rifkin’s Baroque Beatles Book.
Only parts of these songs have counter-melodies. Perhaps you just didn’t notice them*. In “Scarborough Fair,” in the middle verses, Art Garfunkel sings a completely different song (“Canticle”) at the same time Paul Simon sings the main melody.
“Help” is an example of a slightly different thing – a relatively complex background vocal part in the verses (not a completely different song, as in “Scarborough Fair”). Try it yourself! Get together with a friend. You sing “my IN-de-pen…” You have to hold the “ennnnn,” while your friend then sings a different melody/rhythm: “My independence seems to vanish in the haze.” Etc.
*A mark if a successful song, you could say…the counter-melody is so well chosen and integrated that you didn’t feel anything going “against” anything else.
I don’t think anyone mentioned Baby It’s Cold Outside.
Great example! I’d never heard that before.
Here’s the aforementioned All for the Best. Counter-melody section starts around 1:52.
I think of a completely different melody running concurrent with the first (original) melody as countermelody. An example that comes to mind is the third verse of “You Got Your Troubles-I Got Mine” the Fortunes did in I think '65.
Grace Slick, Paul Kantner and Marty Balin were masters of vocal counterpoint with the Airplane.
How about ‘Time of the Season’ by the Zombies?
mmm
I guess it’s not really a ‘pop’ song, and if I understand the concept right
The Battle of Evermore
Offered as much to see if I understand the concept as anything else:
Crosby, Stills & Nash: Teach Your Children Well
Ting Tings: That’s Not My Name
The second one is harder to pick up on but you can see what the man is singing here (“Jules in background”). It’s clearer in the ending of the song as well.
Yes! You beat me to it. This is the example that always comes to mind.
Peter, Paul and Mary withA’Soalin.
Not just the vocals, but the guitar work as well.
Does progressive rock count as pop in your book? If so, Knots by Gentle Giant is prime example.
For more conventional rock/pop stuff, majority of Queen’s output from the 70’s.
Would Balin & Slick’s vocals on “Miracles” be a good example?
Great example, and no doubt the only decent song the “Starship” ever recorded.
Suite Judy Blue Eyes and probably lots of other stuff by CS & N (and sometimes Y)
December by Collective Soul. Or at least the chorus does.
Does anyone else find themselves singing the uh, counter part? I don’t always pick up on them right away but once I do I naturally gravitate to singing that part.
During an in-car performance of “That’s Not My Name”, I sing the female vocals while my wife sings the male counter-vocals 
Good one! I do that too, except I have to switch between the two because I’m solo.
I’m not musically astute enough to know exactly what is meant here, but Blinded by the Light seems like a fit given the other songs mentioned.
The ending of Meatloaf’s Paradise by the Dashboard Light.