As mentioned above, SET is pure class A "by definition", though you can have SET with paralleled output tubes which - from the outside - is hard to distinguish from a push-pull configuration.
Most tube amps are push-pull AB. The reality is that running power tubes in full class A necessitates either drastically shortening tube life or cutting power output way, way down. Neither of which is conducive to sales. And on the flip side, though I have no hard evidence - if "feels" to me that most of the sonic compromise attributed to Class AB solid state (versus Class A SS) is mitigated in the PP tube domain - maybe due to the larger amounts of "benign" low order distortion (masking effect) and/or lower amounds of GNF - contributing to masking the AB crossover distortion (or whatever).
Cathode bias circuits used to be quite common - these tube amps don’t have normal bias adjustments - maybe a "balance" pot at most (to help match the push/pull sides). These amps run hotter per Watt, and much further into class A territory. They’re not so common now; auto-bias circuits (Prima Luna, VAC, ARC) are the new way to achieve cooler running combined with "optimized" performance, and old school cathode bias eats up modern Russian / Chinese power tubes. It was a different story back when we had golden-era tubes aplenty - Mullards / RCA / Tung-Sol / Sylvania / etc - that you could abuse all day (every day) and they’d keep ticking for years. Lots of Mullard power tubes out there with visible horrid scorch markings on their plates, that still measure & play fine lol.
Tube pres and phono stages are pretty much all class A - power is not an issue there. There’s no reason not to do class A.