Some of your picks are awful...
I think essential means songs that represent what the band stood for or what made them popular or important.
Cars: Just What I needed, Shake it Up, Drive
The Police: Roxanne, Message in a Bottle, Every Breath You Take
R.E.M. The One I Love, Orange Crush, Losing My Religion
VH I: Eruption/You Really Got Me, Jump, Panama
VH II: Why Can't This Be Love, When It's Love, Right Now
U2: Sunday Bloody Sunday, Where the Streets Have No Name, The Fly, Beautiful Day (U2 is the greatest band of the last 30 years, kind of hard to pick just three. They probably have about a dozen songs that would be 'essential')
John Mayer (studio albums): Your Body is a Wonderland, Daughters, Waiting on the World to Change
John Mayer (live): Neon (acoustic) Gravity, Belief
Sting: If You Love Somebody Set Them Free, Fragile, Desert Rose
Eric Clapton (solo): Wonderful Tonight (especially live versions) Bad Love, Forever Man
Seal: Crazy, Prayer for the Dying, Love's Divine
I listed John Mayer twice because his life stuff is very different than his studio stuff. His live stuff is very guitar driven. If you haven't seen any of his live stuff I highly recommend it, I was really surprised how good he is live.