I have to admit a bias here because I teach XSLT for a living. But, it might be worth covering off the areas that I see my students working in. They split into three groups generally: publishing, banking and web.
Many of the answers so far could be summarised as "it's no good for creating websites" or "it's nothing like language X". Many tech folks go through their careers with no exposure to functional/declarative languages. When I'm teaching, the experienced Java/VB/C/etc folk are the ones who have issues with the language (variables are variables in the sense of algebra not procedural programming for example). That's many of the people answering here - I've never gotten on with Java but I'm not going to bother to critique the language because of that.
In many circumstances it is an inappropriate tool for creating websites - a general purpose programming language may be better. I often need to take very large XML documents and present them on the web; XSLT makes that trivial. The students I see in this space tend to be processing data sets and presenting them on the web. XSLT is certainly not the only applicable tool in this space. However, many of them are using the DOM to do this and XSLT is certainly less painful.
The banking students I see use a DataPower box in general. This is an XML appliance and it's used to sit between services 'speaking' different XML dialects. Transformation from one XML language to another is almost trivial in XSLT and the number of students attending my courses on this are increasing.
The final set of students I see come from a publishing background (like me). These people tend to have immense documents in XML (believe me, publishing as an industry is getting very into XML - technical publishing has been there for years and trade publishing is getting there now). These documents need to be processing (DocBook to ePub comes to mind here).
Someone above commented that scripts tend to be below 60 lines or they become unwieldy. If it does become unwieldy, the odds are the coder hasn't really got the idea - XSLT is a very different mindset from many other languages. If you don't get the mindset it won't work.
It's certainly not a dying language (the amount of work I get tells me that). Right now, it's a bit 'stuck' until Microsoft finish their (very late) implementation of XSLT 2. But it's still there and seems to be going strong from my viewpoint.