The best screen resolution to design for, is obviously the one that your users use. So to figure out the answer to this question, you would need to perform a survey amongst your users.
For instance, most browsers got the ability to zoom these days, this means that many of the arguments against pixel-based layouts doesn't exist anymore. This however, doesn't mean you should happily use fixed pixel based layouts, without considering the alternatives.
Pixel based layouts are still bad, just for less reasons.
The best screen resolution
The easiest answer is that there is no "best", regardless of what the majority of your users are using. Because what your users are using may change very quickly, and leave you with another pending redesign on your hands. Its therefor best that you think forward, avoid pixel based layouts for websites you expect to be important for a long time.
You could have a look at hollywood movie websites, and different gaming sites. These are often fixed or semi-fixed, but its not as important, because they likely won't stay important as long as other websites.
The best design
Think forward. So what has steadily been increasing over the years? The screen resolution! Most screens these days support full HD, and will likely continue to increase their resolution. Simply pixeling the text size can therefor be bad, using em for text is better.
It doesn't matter much that you use raster-based elements for borders, and other parts on your site, as it doesn't look to bad when done right, not even when viewed in higher resolutions.
Its not so much about what resolution is best to design for, but what techniques that you use. Really skilled designers might even decide to build a background in "layers" using z-index and positioned elements, because that would allow the background to fill the entire browser window in all resolutions.
If you go with a flexible design like the one described above, you will hit some limits as of what is possible. If you go with a fixed layout, you likely risk having to design multiple versions of your layout, one for each the different screen-resolutions.