The following catagories explain what elements a spell counts as. Some spells will be listed in more than one elemental catagory. Any spell listed in X catagory will be able to be refined by an ability of that type, but will not nessisarily count as that element for the purposes of resistance in battle. Check individual spell descriptions for battle effects. Note that some of the spells listed here are not in the standard SeeD matrix. As time goes on, more and more spells will be added, for which you can choose to get new gems, or not. Some will obviously be more useful than others. There will be several new catagories listed here, so browse around and see what sorts of magical goodies are upcoming.

Two additional important bits of info. Spells marked with a * require more than one magic refine ability to refine. For instance, a Storm Elemental spell requires Lightning, Water, and Wind magic refine abilities. Check the full list of elemental catagories, you have to have every refine ability that shows the asterisk marked spell in it, to refine that spell.
Second, when scanning a monster for Spellshards, only shards that you are familier with (i.e. are able to go into your matrix) will be shown. Monsters have LOTS of complex magical weaves inside them, part of discovering spellshards is knowing what to look for. When you aquire new spell matrix gems, you can re-scan monsters you've seen before to see if they carry the new magic that you need.