What are the ideal/typical widths that an adaptive/responsive website should be designed for in order to accomodate the most devices possible?
I design for 320px wide and up. You shouldn't design for a specific set of device sizes because the range of sizes is continuously increasing - a comprehensive list of device sizes isn't comprehensive for very long.
The current trend is to design breakpoints with concern for content, not device widths, and I think this approach will work well going forward, I don't see it obsolescing any time soon.
Below are the sizes I like to design for; not all of these may be ideal for your needs, however I find this tend to provide the cater to the most common configurations of devices out there. When I refer to device width, it is in "device independent pixels" :P
This is the typical device width of 1:1 scale tablet in landscape mode, which also lends itself well to larger computer monitors.
This is the default viewport width for iOS devices (which is supposed to represent the "average" website width), which also lends itself well to average sized computer monitors.
This is the device width of an Apple iPad in portrait mode. Note: Many 16:9 Android tablets have a device width of 720px: Android Developers: Supporting Multiple Screens
This is the typical device width of 1:1 scale phone in landscape mode.
This is the typical device width of 1:1 scale phone in portrait mode.