I have Ubuntu 22.04 on a Dell XPS Plus 13 with OLED display. Looks great, battery life is good. Not sure how tuned the drivers are etc but definitely no need to avoid.
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Battery life is decent on Wayland and not so much on Xorg.
The only issues I can find online are related to brightness control, which has always been an occasional problem on laptops regardless of the display technology. It's happened and keeps happening with LCD panels too.
In a technical sense 99% of the time the kernel is not even involved in the process. The display output is driven by the GPU, and even then, it drives pretty standard protocols like DVI/HDMI/DP. Even laptops typically work with some sort of internal DVI/DP connector for the panel, which has its own decoder made for the actual OLED panel. The only outlier is brightness control which is sometimes done in a weird hacky way by manufacturers, wherever is more convenient for them. Some do it with ACPI calls, some do it through the GPU (intel_brightness), some do it through proprietary registers or I2C/SPI bus.
It's a "try it and see" kind of situation that depends on the precise hardware combination of a given laptop. Being an OLED might make it more likely by manufacturers to end up with a less standard brightness control, but it's not an OLED-specific problem and I doubt we'll ever see a "all OLEDs are now fixed forever". We might see brightness fixed on this particular laptop from this manufacturer.
I wish I had something to contribute to answer this question, but I have not owned an oled monitor to test this on. But I would be curious to see if there is a difference between oled on linux and not. I have seen some people mention poor color accuracy when using amd cards in Linux, but I also have been unable to test this.
Maybe not too late, but System76 has a tool for OLED displays, I'm not sure it works with all displays but probably it worths a try:
I own a zenbook with an OLED display and it runs just fine. Battery life is good too.