* AmigaOS 3.5-3.9. By Haage & Partners. Proprietary. Runs on original 680x0 hardware. Based off the original binaries, with many added components and some bug-fixes & updates.
* Amiga OS 4.x. By Hyperion, originally licensed by Amiga Inc. Proprietary. Based on officially-licensed original sources, rewritten for PowerPC. Runs on some recent licensed PowerPC-based Amigas & some PowerPC accelerators for original machines.
* MorphOS. Proprietary. a 3rd-party recreation, sponsored by Genesi, natively for PowerPC, with rumours of an ARM port. No original sources used. API-compatible with AmigaOS 3.1. Runs on unlicensed, non-official more recent PowerPC machines. Rumours of developers not being paid. No development in a while.
* AROS. FOSS. Runs on x86, with ARM, PowerPC and 680x0 ports available.
All are API compatible with Commodore AmigaOS 3.1. AFAIK, all include emulator(s) so they can run original binaries.
Nothing is very compatible with 3.5-3.9 as they were 3rd party extensions.
None of the successors are very compatible with each other.
AmigaOS 4 and AROS are experimenting with multiple-core support.
Personally, I don't see a lot of future for any of them. The original AmigaOS was something of a triumph, but it achieved its small size and good performance because it targetted the 680000 -- yes, it does full pre-emptive multitasking, but there's no inter-process memory protection, no virtual memory, and poor hygiene between OS and apps or between different apps, making it hard to adapt to more modern hardware. AIUI the communication mechanism between the kernel and apps uses shared memory, meaning that even Commodore itself was unable to enhance it to use the MMU of later 68030-based Amigas, because that would have broken backwards-compatibility.
If anyone fixes that, that will make it less Amiga-like and less compatibile with the old OSes.
I have the demo version of MorphOS running on an Apple Mac mini G4. It works fine and it's fairly quick (until the demo version times out after 20min and slows down.) But there's no Wifi or Bluetooth support.
It looks and feels like, well, a fast Amiga 1200 (the only original hardware I own.) In other words, very retro and a bit clunky. The Mac mini is a machine built to run Mac OS X 10.3 or 10.4, glossy modern 21st century OSes with transparency, scaling, versatile networking and filesystem support, and some high-profile commercial apps. MorphOS feels like a 1980s OS with some bells and whistles glued on.
AROS probably has the brightest outlook and I can see a niche for it as a low-resource graphical OS for low-end ARM machines such as the Raspberry Pi. But the capabilities of such machines are improving fast - the RasPi 3 has 4 x 1-GHz cores, a capable GPU (which need restrictively- proprietary drivers), a gig of RAM, Wifi and Bluetooth. AROS runs under Linux -- the native port isn't finished -- supports 1 core and no wireless. It hasn't caught up with the hardware as the hardware evolves to leave it behind, becoming more capable of doing justice to a modern graphical Linux desktop.