I don’t make a lot of buttons, but do enjoy it when I get a decent one made. For me, buttons are a challenge to make. I like a symmetrical round button and the nature of making a button, makes this more difficult than making a bead.
There are primarily two types of lampwork buttons: shank (glass and metal) and holes. There is another option, a normal one holed bead (usually a disc type) used as a button. Lampwork glass buttons can be used for: closures for sweaters, jewelry, and purses accents on hats/scarves, etc.
Check out this button closure on this felted purse. Nicole Valentine is showing it in her button listings on Etsy to give ideas on how great her buttons look. She has a wonderful selection of beautiful flower buttons. Check them out.
It is not very different than a bead, but a few extra things that I think about when I’m choosing a button to use.
If the button is made on a mandrel, make sure the holes are cleaned of the bead release. When beads and buttons are made on a mandrel, they are made over a thin metal rod (mandrel) coated with a ceramic paste called bead release. This release will break away from the metal and allow the bead/button to slip off the mandrel. This process makes the hole in the bead/button and artisian glass button makers will clean their hole of this powdery substance. Certain buttons are made without a mandrel and therefore will not have the release. See more information on shank buttons below. 
Button by Linda Lawrence of Bella Bead Jewelry on Etsy.
Make sure the holes are large enough to get your stringing material through. Thin Softflex will fit through most holes, but a chunky yarn may not. Button makers should list the size of the holes and if they don’t, ask them, they would be happy to tell you.
Metal Shank Buttons by WildBird on Etsy.
The button should be annealed. Artists will usually mention that their button is annealed, if they don’t, ask and a responsible artist will tell you if they annealed their item. Annealed is the process of soaking the item in a kiln to slowly cool the glass down. This allows the stress in the glass produced during manufacturing to be reduced and decrease the risk of breaking. However, glass is brittle and could always break if the stress is right, so be careful with your buttons (for example, remove them when washing a garment).
Look for raised designs (such as dots and scroll work) to be melted in enough. Just like on beads, the designs should not have an undercut. Undercuts leave less glass-to-glass attachment and could cause the design to break off. If your button going to be getting heavy use choose a design with melted in elements. Check out the wonderful dots on this button by Tera Belinsky-Yoder (aka Beadygirl Beads).
The design of the button should be useful to your project. I like making raised dot buttons (the first button on the post is mine) however, on a garment with loose weave yarn, a raised dot design may get wrapped up in the yarn and be irritating to undo all the time. If this is a concern, choose a smooth round button. Do you want a button to match a whimsical design? If so, look for a button that is in a different shape or character (like a fish, flower).What size do you need? Make sure the artists have accurately measured the button. Will it fit the hole in the fabric/weaving? If the photos are not clear, ask the artist, they should be willing to clarify the measurements for you.
What type of shank does the button have? There are two possibilities: glass and wire. Metal wire shanks are shaped into a U and then plunged into the molten glass during the manufacture or glued on the back with epoxy, or those really cool shanks above. The metal shanks should be smooth and without rough edges (like glass can be). The metal shanks can be different sizes and maybe easier to sew onto a garment.
Metal Shank Bead by karenemlquist on Etsy.
Glass shanks can be made on a mandrel or free formed into a loop without a mandrel. Glass shanks made on mandrels have consistent hole sizes and (for me) it is easier to get similar shapes and sizes of the button and the underlying bead. This type of shank will have the bead release and may have rougher edges than other types of shanks.
Button with bead shank by Genea on Etsy.
Glass loop shanks should have very smooth holes as they were made with a string of molten glass. For me this is the most difficult type to make, but other artists are much better. These holes, again speaking from my experience, may be inconsistent in size and shape. However, they look the most “finished” to me. Other's opinions may vary. Below is a glass shank button made by me. 
Can that bead you bought be used for a button?
Felted purse with disk bead closure from MandaJ on Etsy.
Can I use a button in Jewelry?
This stunning bracelet is by Cassie Donlen. She has it as an example of how to use her buttons. In researching this article, I found her buttons to be the prettiest for me. They are stunning.
