Sadness is not in the poem, rather in the experience of reading or hearing it. Our experiences are not all the same. Let me share mine with you:

I do not know if the following is a poem, but it is poetic and one of the saddest to me. This was read at my sister's funeral

You will lose someone you can’t live without,and your heart will be badly broken, and the bad news is that you never completely get over the loss of your beloved. But this is also the good news. They live forever in your broken heart that doesn’t seal back up. And you come through. It’s like having a broken leg that never heals perfectly—that still hurts when the weather gets cold, but you learn to dance with the limp. -Anne Lamott

Though it seems to bear a message of hope, that limping dance I learn is sad all the same. Each misstep reminds my of that I once had and no longer do. If you do not consider that poetry, then maybe my words below

I tried to run from grief.

At first I thought I could not handle grief, so it manhandled me. I thought I could not be with it, so I did all I could think to do. I ran. I tried to escape. When I could not run fast enough I biked. I could lose myself in the pumping of the pedals, in the rhythm of the road. But as I stopped sweaty and worn, I found myself there with grief close in tow.

I ran to the bottle hoping drink could dull the pain. Try as I might I found it could only dull so much. One night near blackout drunk grief found me and I knew drinking was no escape, so again I ran, but with another problem as falling into the bottle is far easier than getting out, so I struggle with that still.

I ran into a relationship, though one ill advised. When I was with her I could ignore my troubles, but as that relationship fell apart, so too did I. Low as I was before, I found myself lower still.

Tired of running I've turned around. Now I sit with grief. Though I do not languish there, nor stay forever, I let it come and do not run away. Hello there, welcome grief. I see you've come for a while. I am not grief, it does not own me, though I sit with it. So now as I hear happy birthday I still cry recalling her, but they are beautiful tears. Welcome.

If not those, then happy birthday itself, it has a rhyme scheme after all. Though it seems a happy song to most, that last link will explain the experience of it to me.

Sadness is not in the words, it is in the experience. I have tried to share my experience with you so you might see how those are the saddest poems in the world to me.