what about one of those farm-style sinks? Since they sit out further than the countertop, I'd think they'd be wider....

One thing I like about the farm sink is that the front is flush with the base cabinet so no bending over to get at the dishes which is required for most other sinks that are placed further back.

I've had this one marked for our someday new kitchen - it's 24 inches deep.

Here is a link that might be useful: Kindred 24 inch sink

I think you will like this sink ... the interior dimension is 19" front to back and 21.5" left to right. The interior height is 9", and it is undermount, stainless, 16 or 18 gauge, single bowl. It fits in a 24" cabinet. And, it's not very expensive !!
http://www.mrdirectint.com/2421-d-bowl-stainless-steel-kitchen-sink.html

Just in case someone else is searching, here is the same sink in a double bowl ... the large bowl is the same interior dimensions as the above single bowl (19 x 21.5 x 9). The second bowl is a great size for smaller tasks at 16" front to back, 10.5" left to right, 7" deep. You can get it in lefthand or right, 509L or 509R. It takes a 36" cabinet.
http://www.mrdirectint.com/509l-offset-stainless-steel-kitchen-sink.html

This sink configuration is sooooo useful for me. The big side is 19x21.5x9 (23" diagonal) big enough for everything to lay flat in the bottom - cookie sheets, the broiler pan, my largest skillet with the extra long handle & a helper handle on the other side, the enormous canning/crab pot, etc, and its big & deep so the dirty dishes are hidden from view until its time to wash or go in DW. Then I also have the smaller right hand sink that is 16x10.5 perfect for washing dishes, the water fills up fast, & it's even big enough for my 8qt pots to fit in (I find the big side takes forever even to just get an inch or two of water over the grate, but with the double I can use the small side when I don't need the big space). When it's just a few things, I like to wash in the small side and lay out on the sink grid in the large side to drain. If it's more than a few items, they go in one of the dishwashers. And as a bonus ... both of the sink grids fit in the dishwasher ! No scrubbing those grid intersections by hand.

Would you be able to get closer by installing a rectangular sink sideways?

I am happy to find a deeper (depth) sink. I the deepest bowl I can find is 10.5" and I want something a little deeper. My current farm sink is 9" deep and just, well, pretty useless to me.

@fori - Funny you should mention it. That's actually why I'm looking for a sink with a larger front-to-back dimension... It's going to be installed sideways in a kitchen island, so that it's usable from both sides of the island.

@CEFreeman - No offense intended, but please don't hijack this thread with offtopic questions. It's been very difficult to find info on this issue.

@everyone else - thanks for the suggestions! Looks like there are at least a couple of options in stainless, in any case. (Though I'm also still plugging away trying to find something in a composite or other material.)