I have 12 titled “Intro” and 10 “titled” “Untitled” (not to mention Sigur Ros and Kendrick Lamar’s albums filled with untitled songs). I assume those shouldn’t count. 
I could find three titles that are each the name of four separate songs in my collection. A lot of them are album tracks, of course; I’ve bolded the artists of the songs I could hum parts of before listening to them again.
“Animal”: Def Leppard, Miike Snow, Pearl Jam, R.E.M.
“Gone”: Ben Folds, The Beta Band, Pearl Jam, U2
“Run”: Gnarls Barkley (released as a single without its subtitle, “I’m a Natural Disaster”), New Order, Snow Patrol, Tin Machine
Three different songs with multi-word titles:
“All I Want”: LCD Soundsystem (2 versions), Passion Pit, Toad the Wet Sprocket
“I Want You”: The Beatles, Bob Dylan, Rachael Yamagata
“Run For Your Life”: The Beatles, Big Grams, DJ Shadow
“4th of July”: Aimee Mann (2 versions), Soundgarden, U2
A bunch of one-word song titles I have three songs of:
“Congratulations”: Dessa, Juliana Hatfield, Paul Simon
“Friends”: Flight of the Conchords, Led Zeppelin, The Police
“Girl”: The Beatles, Beck (2 versions), Jamie xx
“Helpless”: CSNY, Soo/Miranda and Ashanti/Ja Rule (same song), Sugar
“Higher”: The Cardigans, Carly Rae Jepsen, The Naked and Famous
“Low”: Coldplay, Foo Fighters, R.E.M.
“The Ocean”: Led Zeppelin, Tegan and Sara, U2
“Quiet”: The Beta Band, Rachael Yamagata, Smashing Pumpkins
“Reunion”: M83, Stars, The xx
“Run”: New Order, Snow Patrol, Tin Machine
“Tomorrow”: Avril Lavigne, Thievery Corporation, U2
“Trouble”: Coldplay, Shawn Colvin, TV on the Radio
“Wait”: The Beatles, M83, Sarah McLachlan
“You”: R.E.M., Radiohead, TV on the Radio
The sets of three really popular songs with the same title I can think of are “Creep”, “Drive”, and “One”, but I’m missing the TLC, Cars, Incubus, and Metallica songs.