What is the 'most common' width and height that websites are being designed for these days?

Like you said there is no standard height as that is extremely fluid so just use whatever helps you visualize your design best.

For width 960 pixels wide seems to be current "standard" for fixed width designs. It's common enough that a website has been dedicated to exploring layouts at that width.

(I'm assuming you don't want anyone ranting about using a fluid design?)

There really isn't a standard width you should be setting. The amount of people using a 1024x768 monitor has dropped drastically in the past few years, so a width of 960 has less importance. With the emergence of so many new devices (smart phones, tablets along with laptops and desktops), you should create a number of stylesheets for each case.

CSS media queries allow you to change styles based on the user's screen resolution. I tend to make sites with a general stylesheet that handles all of the visual styling, then separate ones for mobile, tablet, standard desktop/laptop and extra wide screens that structures the page.

Check out this page on W3.org for a few examples - http://www.w3.org/TR/css3-mediaqueries/

It is more work, but will allow the user to get the best possible experience no matter what they use to browser their site.