What can you do with Lightroom catalogs if Adobe shuts down support for Lightroom? Is there any other software compatible with Lightroom catalogs? I don't have Lightroom, this is one of the things that will make my decision.
I think you'll be fine. Look at the case of Aperture, discontinued by Apple. Adobe offers an Aperture import plug-in.
Lightroom's catalog is an SQLite database and much of the data stored within it is in XML format so the data is very accessible to programmers if the need arises. In the unlikely event that Adobe and Lightroom ceased to exist, it would easy enough to retrieve data from a Lightroom catalog.
If Adobe stops support for Lightroom, you use your catalogs just as you did before. You don't need Adobe's support to use the software, especially if you have a perpetual license. But even if not, the "read only" features will still work after Adobe revokes your CC license.
Other than that, there is no migration path to other software currently available. There may emerge limited tools to migrate tags etc. if the need arises, but develop adjustments won't be preservable, other than by export to TIFF, just like it was with the recent end of Aperture.
Lightroom is a very successful product. I suggest you not worry too much about this possibility until it happens.
It's fairly safe to assume if Adobe does decide to abandon Lightroom, someone will take up the opportunity to create an importer that will suck up your Lightroom data into their product, the way Adobe did with Apple Aperture recently.
To be honest, of the issues that might or might not cause me to decide about starting to use Lightroom, this one is in the ten or so closest to the bottom of the list. I personally wouldn't let it affect my decision.
The description information in the LR's catalog (like keywords, rating, title, etc.) can be easily written into your images as XMP metadata. And any decent DAM solution that supports XMP/MWG can easily import it.
Information about your non-destructive RAW adjustments also can be written into the XMP but this is unuseful for other programs because these are just arguments of the proprietary functions that nonone except Adobe might interprets.
It's not unheard of for companies to drop support, or to be bought out and killed off, or to become so costly to maintain for you that you decide to migrate away from it.
There's not really a way to combat this, and Adobe does have a good history of maintaining legacy support.
Even so, you are taking a risk by putting your work entirely into the hands of a closed source company. If you want to own your data, metadata, and the very file formats it is all stored in, then you should check out open source solutions. While the source code of these applications may do you very little good (unless you happen to programme), it would mean that all other users of that software who DO code can ensure continued support or migration of the format.
Open source solutions are unique applications unto themselves. They do not seek to just copy similar closed source apps, so there would be a learning curve. For some people, however, it's worth it. Was for me.
I agree with most that you will not have to worry for some time that Lightroom will stop being developed and even if it does stop being developed, it will be able to be used for some time afterwards (since an internet connection is not required to run, only update). The problem with running software that is no longer maintained is that computer equipment gets old, operating systems update and Lightroom may become unstable or uninstallable on future machines. This is called software rot.
Nonetheless, there is an open source product that has a strong, but not perfect, import for Lightroom. The product is called Darktable and there is a blog post on importing a Lightroom library here: http://www.darktable.org/2013/02/importing-lightroom-development/. Pascal, the engineer, posted in Feb 2013 but he was using Lightroom 4, which is great to see.
You can also be almost guaranteed that there will be an increase in quality and quantity in the engineers that help with that product, or others, if there is a gap to be filled. If Lightroom goes down, Darktable and others will do whatever they can to fill the void.