Two of my favorite singing and flatpicking songs were made famous by Ola Belle Reed - "I've Endured" and "High on a Mountain". Tim O'Brien and Hot Rize covered these too.
Any flatpicking song can be as difficult as you want to make it, but with a few major pentatonic runs (aka G run, C run), some hammer ons and 7ths you can make it recognizable and do it justice.
Some more of my favorites:
Been All Around this World (Foghorn String Band or Hot Rize version)
Georgia on a Fast Train (Billy Joe Shaver)
Dark Hollow
I've Lived a Lot in my Time (bluegrass version, NOT Jim Reeves)
Grandpa Was a Carpenter and Please Don't Bury Me (John Prine)
And lots more. You can do a lot of old country and folk songs in a flatpicking style. Just about every note you need to pick the melody is right there in the chord shape you're holding. I'll frequently pick out the melody while keeping the beat at the same time. It might sound hard but if you keep at it, it becomes intuitive.