Set your expectations first
If you searched for an under-cabinet wine rack expecting to hang bottles under a kitchen cabinet, this page will save you some time: that is not really what this category is, and it is not a gap in the market so much as a sensible absence.
The space under a wall cabinet is shallow, and a wine bottle is heavy and awkward. Suspending several of them overhead, over a worktop where people stand and work, is a bad idea for reasons that have nothing to do with wine. So the products that exist in this space hold stemware, hanging glasses upside down by the base of the stem.
Which is genuinely useful, just differently useful. If you want bottles off the counter and out of the way, look at wall-mounted racks instead, where the load goes into a wall rather than into the underside of a cabinet.
Glasses, or glasses and a decanter
The Wine Enthusiast Large Under Cabinet Stemware Rack is the straightforward version: a rack that screws to the underside of a cabinet and holds glasses by the stem, out of the cupboard and out of the dust.
The Oenophilia Fusion Stemware & Decanter Rack takes sixteen glasses and adds something most racks do not: somewhere to invert a decanter to dry. If you own a decanter you already know this is the annoying part — a wet decanter is awkward to dry and worse to put away damp. Having a dedicated place for it is a small thing that gets used constantly. If you are shopping for the decanter itself, see best wine decanters.
Neither listing publishes a load rating, and neither needs to carry much — but the fixing still goes into the underside of a cabinet, which is usually thinner material than a wall. Use screws of the right length for that material: long enough to hold, short enough not to come through the shelf inside.
Where this genuinely shines
In a butler’s pantry or a drinks cabinet, where the whole serving operation can live in one place. Stemware hanging under the cabinet, the decanter drying beside it, bottles in a rack or a cabinet below. Everything for pouring wine within arm’s reach, and nothing on the counter.
That layout is also the easiest built-in wine fridge installation there is, if the space eventually earns one — see built-in wine fridge ideas.