Sharon that's a great question. I guess it's hard to know for sure unless you have little kids or grandkids around to ask!
It's reassuring to read Dusty's reply, that many of the same songs are still taught today in school. 
If that's so, then you might find some good ideas in this old forum thread:
music-discussions/what-songs-were-you-taught-in-kindergarten-grade-school
It's mostly the same tunes, but with updated lyrics. Instead of "If you're happy and you know it clap your hands" you have to sing "If you're happy and you know retweet this meme." 
Seriously, my kid was in kindergarten about a decade ago, and the songs were basically the same as when I was a kid. In fact, you're giving me a good idea for a "Call the Tune" for any songs for kids. We could all share our ideas for good songs for kids.
@Sharon-Day Great question! I know what songs I've sung to/with a little niece or nephew yet have no clue what songs kindergarteners today know.
I’m going to be teaching mountain dulcimer at our library’s Kids Fair? Do today’s kids still know Aunt Rhody? Twinkle, twinkle? Mary had a Little Lamb? My question is what songs do doday’s Kindergarteners (and beyond) sing? I’d like to teach them to play songs that they know.
