Well the standard model doesn’t predict them, and apart from one detection that has never been reproduced we’ve never detected any pre-existing ones or any created in a particle accelerator.
Does the SM theorize about the impossibility of the monopoles' existence?
I have not heard anything about the SM predicting them being impossible, but I assume it does given that I haven't heard anything about the subject. Source: I have a master's in physics and a friend of mine did his bachelor's thesis in monopoles in classical electrodynamics (albeit modified) and his master's in quantum ones, so he would've told me if he knew.
However, if any GUT theory we have come up with is correct (because a few exist), then that does imply the existence of monopoles.