You've recieved several answers to this question in this thread. However, I don't think any of them are nearly complete, and might even be misleading.
Condensed matter scientists make up a large group of all physicists, and I've read reports that suggested condensed matter and materials physicists make up more than two thirds of all physicists. No one has mentioned anything that they work on in this thread; neither have they mentioned modern studies in decoherence, electromagnetism, quantum logic or other useful areas of physics. Basically, almost no scientists are searching for anything that has been listed here.
One great way to see what physicists are exploring is to go to university web pages and look to see what research is being done. You'll find that only the biggest universities have the time and money they can spend on high energy physics, and even then this is only a minority of physicists at the facility.
Here is my take on one of the biggest questions in physics right now (which, of course, is composed of myriad smaller questions):
How can we best predict the properties that emerge when matter is condensed and/or strongly correlated and placed in various conditions?