Faggoting lace is the most basic kind of lace there is. Or, as Barbara G. Walker put it in A Treasury of Knitting Patterns:
It is, in fact, lace reduced to its two bare essentials: (1) a yarn-over stitch and (2) a decrease. Typically, faggoting consists of just these two operations ant the pattern had only one row. In spite of this simple structure there are a surprising number of different types of faggoting.
The differences lie in what type of decrease is used, whether the decrease comes before or after the yarn over and whether there are any plain knit or purl stitches between the yarn over and decrease sections.
All of these patterns produce a very stretchy openwork pattern that is great in panels or as an allover pattern and is particularly great for socks or fancy gloves.
Turkish Stitch is one of the most basic forms of faggoting, consisting of yarn overs and knit 2 together with knit stitches on each end of the row. Specifically, the repeat, which works on an even number of stitches, is:
K1, *yo, k2tog. Repeat from * across, ending with a K1.
Repeat this row.
An alternative version uses ssk instead of k2tog for the decrease, which does give it quite a different look.
Another alternative is Purse Stitch, which uses purl 2 together as its decrease.
Garter Stitch Faggot uses the same pattern row as Turkish Stitch, but it is alternated with a row of Garter Stitch, which makes the fabric a little firmer and more solid (it also biases thanks to all those right-leaning decreases; you could alternate a row of k2togs with a row of ssks to reduce this tendency).
These stitches can look very different depending on the weight of yarn you use, the size of needles you choose and whether they are blocked or unblocked. Blocking is necessary to open up the yarn overs, but the look of the fabric is really interesting when it isn't stretched, too.
Patterns Using Faggoting Lace
I used Turkish Stitch to knit a clingy openwork tank top/swimsuit cover.
Crystal Palace Yarns also has a scarf using this stitch pattern, which is a great spring and summer scarf.
Rex Decker's King Cake Faggot Stitch Scarf shows what the stitch looks like when not heavily blocked, which adds cool texture to the colorful yarn he chose.
Faggoting is used as part of the leg for the Faggoting Ribs Socks from Claudia Tietze, which also feature ruffles, a lace panel, and cool gusset shaping. (It's a free pattern, but you have to register to download.)
If you're willing to pay for a pattern, these Steampunk Faggoted Rib Mitts from the Little Songbird are cute and would be great for wearing with a costume or just every day to take the chill off in a feminine way. The Faggoting and Beehive Lace Tee by Amy McElwain is another paid pattern that uses openwork stitches for the bottom part of the shirt.
And if you happen to have the book Together or Separate by Ann McCauley, there you will find a Moss and Feather Faggot Shell, which uses the faggoting on the yoke and Moss Stitch on the body of the top.
Faggoting lace patterns are a lot of fun to knit, and another alternative would be to make your sampler of the stitches. Many of the basic patterns work on any even number of stitches or multiples of 4, so you can work a scarf with, say, 16 stitches and do a foot or so of scarf in each of four or five different stitches.
You'd learn a lot about how increases and decreases work and have a great scarf at the end.