Are Swing applications really used nowadays? I don\'t find a place where they are used. Is it okay to skip the AWT and Swing package (I learned a bit of the basics though)?
You should certainly write Swing for desktop Java, only using AWT to the degree that Swing demands it.
I'm sure people can come up with examples of very good Swing apps besides SQL Squirrel and IntelliJ.
I think that RIA technologies like Flex, Silverlight, etc. are ascendant because the web and services are taking mindshare from desktops. It's not just Swing that's losing out.
There is no universal question to whether it is ok to skip something or not. It's a matter of priorities. While Swing has a lot to teach you (and it is too heavily infused with AWT), if you're never going to be writing GUIs you may be better off investing in something else.
Swing, while ugly as hell, is still used in many places. A quick search on "java swing" in Indeed or dice would reveal many jobs and industries that require it.
Swing applications are used in most cases where a Java app runs on the desktop. Sometimes you don't even know - for example, LimeWire is a Java Swing application.
When learning Swing, you'll find that you will come to know those parts of AWT that are still important, such as Event, LayoutManager, Graphics, Font, Color, etc.
Swing is heavily used in business specific (vertical)/internal application development.
While Swing has some cross-platform look-and-feel deficiencies and it would be nice if it was updated to support generics, it is still well designed and quite usable. I have always been a fan of the sheer amount of customization offered. SWT is nice and is more native, but you give up some power and control with that as well. In addition, it seems simpler to use, which can be a plus, since Swing can be overly complex due to its flexibility.
Geertjan Wielenga often blogs about applications based on the NetBeans platform, and therefore Swing. The applications range across the board from rather small esoteric applications to very large, mission-critical systems (Boeing Aircraft comes to mind). Many of the enterprise-level applications that Geertjan misses are mentioned on Planet NetBeans. It seems like every few weeks there is an announcement or description of such a program.
As noted above, lots of tools use Swing including the IntelliJ IDEA IDE, a favorite. I'm surprised no one else mentioned it.
Sun also used to maintain a Swing Sightings web site, but it hasn't been updated in a few years.
Where I work, Swing is used in the presentation layer for a very large document management system. In my own development experience I've used Swing for desktop applications in the areas of image processing, device control (video, motors, etc.), math, statistics, and medical devices.
BTW, I agree about the slowness of file choosers. But there are lots of alternatives that will use the native file chooser. It can just be a bit jarring if the L&F's don't match up very well.
EDIT: Oh, and as others have mentioned, it's very easy to use Swing from other languages on the JVM. With Clojure, for example, it is even easier to use than in Java, in my opinion.