Working with delicate fabrics can be difficult, as they tend to fray. This fraying can make the fabric unusable, so taking steps to limit any fraying is essential. Depending on how likely a particular fabric is to fray, you have a variety of options you can use for this purpose.
Method of Cutting
If you are using a delicate fabric that doesn't usually fray very much, you can cut the edges with a rotary cutter or pinking shears. Another option is to clip a small triangle off of each corner. These methods may not be enough to prevent all fraying, but they can help minimize it.
Sewing
For delicate fabrics that fray a bit more, you can sew any raw edges to minimize fraying before you prewash the fabric. A straight stitch will help, but using the serge function on your sewing machine or utilizing a zigzag stitch will work better for fabrics that are particularly prone to fraying.
Sealants
There are a number of chemical products made to prevent fraying. These will usually hold up for a number of washes as long as you use the delicate cycle. If you are going to finish the edges of the fabric later, you can use water-soluble glue to temporarily keep the edges from getting frayed until you have a chance to deal with them. The glue will wash off, so you don't have to worry about how it looks.
Considerations
The way you wash your fabrics can contribute to or minimize fraying. For the best results, hand wash your fabrics or use the delicate cycle. The less agitation there is, the less likely your fabrics will fray. However, if you are making something from the fabric that will be regularly washed, you need to prewash it using the same settings you will use for the finished garment to control for any shrinking.