Elastic and lace should be small, so they are in scale with your dolls. For smaller dolls, beading thread may be just the thing. It is easy to zig-zag beading thread along the edge of knit fabrics—it will provide adequate stretch for doll panties or other pull-on garments.
Try to find regular elastic that is no more then 1/4-inch wide. Knitted elastic is easier to work with than braided elastic, although it can be harder to find. It is difficult to sew into braided elastic and you may have trouble. Knitted elastic is easier to sew into and seems to stretch better. For larger dolls, you could possibly use clear elastic. However, if you have a problem, you can only sew over clear elastic once.
Lace should also be small, as well as the motifs on the lace. Cotton lace is best and works well with small doll clothes, although you can use nylon or polyester if you can find nothing else, at least for modern doll clothes. For antique or reproduction antique dolls, you must persist and try to find cotton lace as modern synthetics just don't look right for a nineteenth- or early twentieth-century garment. You have more leeway in the size of lace you can buy - it really depends on what sort of garment you are making and what looks appropriate.
Remember! Keep it in scale. If it looks awkward, you should probably look for something else.
Quarter-inch lace seems to be getting harder and harder to find, but sometimes you get lucky and find it in a fabric shop. Otherwise, you will have to find some on the internet. This is where I do most of my shopping for doll notions, in any case. And the selection is much wider.
Again, the idea is to have as little bulk as possible in doll clothes so you maintain the scale of the small garments!
Beading thread--the kind you use to make stretchy bracelets--is great for smaller dolls' clothes. Use an edging foot to attach it. Hold it close to the edge you want to elasticize and zig-zag over the edge and the beading thread, being careful not to catch the thread with the zipper. The stretchy edge that results will be hardly visible, but will still stretch to fit on your doll!
For larger dolls, quarter-inch elastics may be a good size. But I wouldn't go any larger unless your doll is a 60 cm BJD. To attach this regular elastic, sew it on the way you would any elastic. You can make a casing and thread the elastic through the casing. Or you can attach it by zigzagging over it. Clear elastic is attached in the same way.
All elastic and lace, and other trims should be kept to the scale of your doll.