Way, way back when, I looked at the notable lines from the first third or so of the poems that I'd read. My theory was that one of the ways to tell a quality poem is that a line from it had entered the public consciousness to the point where a quotation would be well understood. This is, of course, subjective to my own experience but that's the only one I have to work with. I said I'd go back and do more but I don't think I ever did. Let me fix that now.
Ode to Joy by von Schiller - nothing jumps out at me here.
Lay of the Last Minstrel by Walter Scott - nothing here.
Kubla Khan by Coleridge - The opening: 'In Xanadu did Kubla Khan/A stately pleasure dome decree...'
Visit from St Nicholas by Clark - virtually the entire thing though the end 'Happy Christmas to all/And to all a good night!' is probably the best remembered.
Abou Ben Adhem by Hunt - no, but it should be.
She Walks in Beauty by Byron - The opening 'She walks in beauty like the night' is fairly well known.
Ozymandis by Shelly - The most remembered is 'Look on my works, ye Mighty, and despair' though the whole thing is fairly well known I think.
Thanatopsis by Bryant - nothing in this one.
Ode on a Grecian Urn by Keats - 'Beauty is truth, truth beauty' is well known.
Concord Hymn by Emerson - Quite fitting to the subject at hand, the phrase 'shot heard round the world' is universally known.
How Do I Love Thee? by Barrett Browning - I'm guessing even young school children have heard 'How do I love thee?/Let me count the ways'.
Paul Revere's Ride by Longfellow - Here we have 'One if by land, two if by sea'.
Barbara Frietchie by Whittier - Fine poem, but no phrases that stand out to me.
El Desdichado by Nerval - Nothing that I recognize but I wouldn't be surprised if phrases here are quoted in his native France.
The Raven by Poe - "Quoth the Raven, 'Nevermore!'". Yep, that's stood the test of time.
Ulysses by Tennyson - The line that stays for me is 'for my purpose holds/To sail beyond the sunset, and the baths/Of all the western stars until I die.' This is undoubtedly due to my Heinlein influence though. What a great poem!
Old Ironsides by Holmes - No, nothing that stand out.
The Owl and the Pussycat by Lear - I think the whole thing is fairly well known but the closing 'They danced by the light of the moon.' is probably the best known.
Of those twenty, thirteen added some line that would be fairly well known. Again, there may be some translation issues here. Or just ignorance on my part.
I don't mean to slight the other poems at all. Some of them are very good, but without that one strong poetic turn of phrase. There's no shame in this, certainly. Poems are built for many reasons. Those that tell a strong story or have a special moral to them aren't bad simply because they didn't have some small portion that took off.