This list should not be regarded as comprehensive (how could it ever be), but here are some of the worthy outlets in which you might seek publication for your own work. Please also subscribe to some of them. Keep the business (in which you hope to find readers) alive.
Acumen, Ambit, Envoi, The Frogmore Papers, Iota, Kaffeeklatsch, Magma, Mslexia, Neon, New Walk Magazine, Obsessed with Pipework, Other Poetry, P N Review, Poetry London, Poetry Review, Poetry Scotland, Poetry Wales, Poetry Ireland, Shearsman, Stand, Staple, The Dark Horse, The Harlequin, The Interpreter's House, The Reader, The Rialto, The London Magazine, The SHOp, The North, The Spectator, H Q Magazine (The Haiku Quarterly), Poetry Salzburg, The London Review of Books, Tears in the Fence, The Times Literary Supplement.
The National Poetry library maintains a list of poetry magazines.
Poetry magazines.org is also useful for contents of selected magazine back issues.
Publication in some magazines carries greater kudos than others. How can you tell which are the best, given that I'm not going to get myself into trouble by telling you here?
Try sending for some of them and check out the contributors (in the biographies, look at their track record - do they have collections to their names? prizes? are they well known?). Check whether the magazine pays contributors in cash or in kind (or both, like Poetry Review). The harder it is to get work accepted, the higher the kudos of the publication. And the ones that pay, tend to be the most highly rated, or at least the most highly rating by arts funders, which is (and isn't) the same thing.
Don't be put off by the fact that many of these magazines receive vast quantities of poems each year. Of the vast quantities they receive, the number of good poems are not vast. So all you have to do, is make sure your poems are good.