I find reasonable results with rod inductors for power filtering; but I don't mind about EMI in an optimized design, I'm just putting the thing in a box anyway or whatever. You never see them in that position, in commercial designs, and I think that's mostly for EMI reasons. For low level filtering however, they're quite popular -- taking the edge off a flyback output, for example.
Commercially, barrel/spool/bobbin/however you want to call them, are maybe more popular (in the larger sizes that aren't practical to support in SMT types). The shorter airgap path performs better, for the aforementioned reasons.
If you invent a wire material with 10x higher conductivity than copper then perhaps a rod core might make sense, but so far we don't have any such wire materials that are cheep or work at room temperature.
It's kind of fascinating that the best conductors we have (at room temperature, at this time) are best suited to a mu_r around 20. The suitability is simply a matter of resistivity versus inductivity, and if we had just a few times better conductor, we wouldn't be very bothered about using cores at all, a lot of the time. It's not a huge difference: it's like comparing the thermal conductivity of aluminum and copper, or copper and diamond. Sadly, there's no "diamond" of electrical resistance at room temperature.
Now if we get superconductors involved, fields can get quite intense. Pencil-fine wires carrying hundreds, even thousands of amperes. Thousands of turns of that, on a solenoid or whatever, and you can store just... a lot of energy. Up to 10, even 20T, given suitable cooling (maybe not HTS, that needs LHe I think). 20T corresponds to an energy density of 160 J/cm^3, or a pressure of 23k PSI (160 MPa) -- at these levels, coilguns become practical for example, perhaps competitive even!
IIRC, the plasma magnets in ITER will (are?) total some
henries of inductance.
Apparently a few short-term electrical grid storage systems are operating now, using superconducting storage.
Tim