I recently learned about these technologies and the differences between them. Which one of these do modern DSLR cameras (esp. Canon and Nikon models) come with?
CMOS sensors almost all use a rolling shutter, CCDs are generally necessary to get a global shutter. It has to do with the way the sensor is read. A CMOS sensor by design has to be read in series and thus creates a rolling effect as values are sampled. The main way that a few specialized sensors get around this is by including memory with each pixel to capture the value so it can then be read sequentially. CCDs on the other hand dump all their information in to a buffer at the same time and the buffer is then read, thus the sampling occurs at the same moment.
Recent advances in CMOS technology has greatly reduced the extent of the rolling shutter on higher end sensors, but it is none the less still present and is not possible to eliminate entirely. The reason CCDs aren't always used is that they are much more expensive, particularly for high resolution applications and don't offer significant advantages for still photography.
Now that CMOS has come as far as it has, it is appearing in lower cost professional video cameras such as the Canon Cinema EOS line while the higher end video gear uses 3 CCDs (one for each color, red, green and blue).
CCDs are not used generally used in still cameras because of the resolution issues. HD video is only about 2 megapixels. Making a color CCD at something like 18+ megapixel resolution is quite expensive. For example, an 11 megapixel CCD camera that I found used was running for $2600 without lens. Such cameras are often only used for industrial applications where both resolution and global shutter are needed.
Modern DSLR cameras come with rolling shutter. My Nikon has visible rolling shutter (in certain conditions).
I don't know whether exceptions exist.
Yes, this site says that CMOS have a rolling shutter and CCD uses total shutter.
However, you should consider whether rolling shutter is a problem for you. For me it is not. You can work around it in most situations, you can also remove some of it in post (premiere pro, apparently). But if you are concerned about this for video shooting, you should ask on http://avp.stackexchange.com.