Nancy Marchand is the undisputed queen of Brioche Stitch. Although many people come to Brioche Stitch through Elizabeth Zimmerman, who called it "Prime Rib." Unfortunately, many people find that Zimmerman's instructions were a little too "pithy" when it came to Prime Rib. I know for a fact I'm not the only person who gave up on that pattern in frustration!
For some reason, Brioche Stitch has been poorly served by various stitch dictionaries. One problem is that there are a lot of different stitches that people call "Brioche Stitch." In some of them, you pick up and knit the stitch below. In others, you slip and then knit different combinations of stitches. I have seen Brioche-like stitches under the name Fisherman's Rib and Shaker Rib, among others.
Marchand has an entire website dedicated to the topic of Brioche Stitch, as well as a number of different books and patterns. You can knit a two-color version of Brioche Stitch in which the colors alternate in vertical stripes - it's an easy trick, and it looks terribly clever! Marchand has used this to good effect in many of her patterns. In others, she handles the decrease and increase lines exceptionally well. A friend of mine knit Marchand's "Newmarket Sweater," and I always admire it when I see it in person!
One quirk of Brioche Stitch is that it uses quite a lot of yarn. Not so surprising, given the thick squooshy fabric that it creates. But definitely something to keep in mind.
For a basic Brioche Stitch scarf, start by casting on an even number of stitches. (I'm working on one that started with 26 stitches in Lion Brand Wool-Ease on #8 needles.)
There are only two stitches you need to master in order to make this basic scarf:
yfsl1yo is Marchand's abbreviation for a move where you move your yarn forward (towards you), slip the next stitch purlwise, and add a yarnover. This is easy to do in one smooth motion, once you get the hang of it.
Brk1 Knit that stitch together with its paired yarnover.
Start with a set-up row: [yfsl1yo, k1] repeat across.
Then repeat endlessly: [yfsl1yo, brk1] until you run out of yarn. Easy peasy!