When you have both a PC and a laptop where is best to store the .LRCAT file?

The problem is always capacity and bandwidth. I use file synchronisation software to keep two desktop machines fully backed up against each other, so that I can work on either at any time. That's the least problem.

The problem is how to transport the files. That's the bit you need to figure out one way or the other. I work with big files and lots of them, so cloud solutions are out. If you can find a way to separate out and move only new and modified files - this will include the entire .lrcat file every time - you can manage with a small pocket size SSD external.

My scheme involves bringing the whole archive on two 8TB disks back and forth, in a separate compartment in my camera rucksack, which I'm carrying anyway. These files are just for transport - when I connect to the desktop the software copies new and modified automatically, and I'm good to go. Plus I have a third backup.

If I could find a way to replace the old files on the externals with placeholders that the software recognizes, it would be ideal. But I haven't found it yet.

Not sure I would recommend this rather elaborate procedure, but it has served me well for many years - and I sleep at night.

Depending on how often you switch between computers, storing it on an external hard drive can be the best solution. Backing it up to the NAS would make sense. It can be a bit annoying to drag around the hard drive, but in some cases it makes sense.

Again, depending on your needs, another possibility is to store the main catalogue (always have a "main" catalogue) on your main computer and then rely on LR's sync feature to give the laptop access to the images and work done on them. So you might have the 580k catalogued in the file on your desktop PC and sync some files to Adobe (eg recent ones, ones you're working on, even everything). You can then access them in LrMobile on your phone, or on the laptop using a browser (LrWeb) or by installing Cloudy Lightroom, and do certain work (adjust, title, caption, rate and flag, but not keywords) which will sync automatically back to the main catalogue. New photos might be imported into either computer, but sometimes you might want to import them into your laptop's Classic catalogue (which you can backup to the NAS), work on them, and finally get them over to the main catalogue which can be done via a few methods depending on circumstances like quantity or network speed.

Lightroom Classic will not run from  network storage by design. It is a single user application only. I think it's best to maintain two separate catalogs (.lrcat). Pick which machine you want to be the principle Lightroom Classic computer. Additionally, Lightroom Classic can only sync to the cloud one catalog at a time.

If you need to have certain critical images on each computer, my solution is to create collections of those  images and sync those images to the cloud. You can then run Lightroom (cloud app) on the other machine to see them and make adjustments. The limitation to this scheme is that the images uploaded to the cloud from the main machine are only smart previews about 2500 pixels in the long dimension.

Over the years, I have heard of some users who have put the catalog on an external drive so it can be connected to one computer and then another. That external drive would have to have an extremely fast connection to the computer to have satisfactory performance. Perhaps someone who has done this will add to this discussion.

Got an indirect answer. I tried loading the .LRCAT file on the NAS (Server) and tried to direct Lightroom to use the file from there but the Lightroom software gave a message that it cannot wotk with files on a network.