Like most foodies who have their own herb garden I always have a good amount of herb stems and I thought I would share one idea on how I use mine. This is a community wiki so please feel free to add your own ideas to this wiki. Just click edit and add your ideas of photos or whatever.
[h2]Left-Over Herb Stem Ideas[/h2]
Herb Oil

I simply take the stems I have left over and place them in a canning jar or any clean jar with a tight fighting lid and add oil. You can add whatever you like and in the oil I am making here I added parsley stems, basil stems, thyme, chives and one clove of garlic. I think let this sit in the fridge for a few days then I use this oil for Sautéing or grilling. It is a great alternative to plain oil and it also adds some flavor to what your Sautéing. Here are a few photos of the steps I did.

* Note, this technique requires you use the oil in a short span of time (one week) and you must keep it refrigerated. We used to use this technique in the restaurants I worked since we would use the oil up usually in a few days.

You can also get somewhat more flavor out of the stems if you blanch and shock them (dip in rapidly-boiling water for 30 seconds, then drop immediately into lots of ice water), then puree them with a small amount of neutral vegetable oil and the water that remains on the cold stems. Then mix the puree with decent olive oil and let stand overnight. This method generally produces a vivid green oil, intensely flavored, which you may wish to strain fine before using, depending on the application. You can use neutral vegetable oil for all of it, which is probably best if you're going to use it with high heat. Note that extra-virgin olive oil should not be pureed in a processor, because it will oxidize and taste bitter. This method is not superior to the above, but produces a different, equally valuable result.
[table]Making Herb Oil Photos[tr][td]Fresh picked basil leaves[/td][td]What to do with Basil Stems?[/td][td]A little fresh time[/td][td]A little fresh chives and parsley stems[/td][/tr][tr][td]
[/td][/tr][tr][td] [/td][td] [/td][td] [/td][td] [/td][/tr][tr][td]Put this into the jar[/td][td]Notice the garlic clove on the bottom.[/td][td]Add some oil.[/td][td]Post the date and into the fridge.[/td][/tr][tr][td]
Herbal Vinegars

Use them to make herbal vinegars. You can do this either cold or by warming the vinegar first. Basically, the same approach: put the herb stems in a jar, cover with the vinegar, and you're good to go after letting it sit a couple of weeks.

Similar to oil, you can make herb vinegar. This product is safe for longer term storage because of the acidity of the vinegar.

Rinse and dry the herbs, about a cups worth. Bring about 2 cups vinegar to a simmer and pour it over the herbs in a clean jar. Cap the jar and let stand for 10 days, shaking the jar occasionally. Pour/strain the vinegar in to another clean/sterile jar. Store in a cool dark place.

Which vinegar to use is a personal choice, though red wine vinegar is a good choice if you intend to use the vinegar in marinades or with added garlic.  The flavor of more delicate herbs comes through better in a rice or white wine vinegar.

BTW, herbal oils should not sit around for any length of time. Store them in the fridge, and use them relatively quickly.

Grilling With Herb Stems

Another use: When grilling, drop the stems on the coals, right under the food. Aromatic herbs in particular, lend a suble flavor as they smoke the food.

Stocks and Sauces

Herb stems can be an excellent substitute for a bouquet garni when making stocks and sauces. As a rule, use only the same kinds of herbs as you would normally include anyway: rosemary, for example, will strongly flavor a stock, and is probably best used in small quantities and in specific sauce applications, such as a base for a lamb sauce. Unfortunately, it is far more difficult to estimate how much flavor will be imparted by stems alone, so you will need to experiment.