Say I spent my points in a manner I now regret, is it possible to reset and respend the points for a particular multiplayer class?
You can't respec in multiplayer, not at whim (you need a rare item, as mentioned by Scott's answer), but you have a different option available to you: Promotion.
This is a purely optional choice, but it is the most easily available way you can reallocate your skill points, since there's no other way to delete your character once made.
The other option, of course, is to simply play a different version of that class. If you're playing as a human, you can create a human of the opposite gender and start your power build from scratch. If things get really bad, you can even switch to one of the other races, level them up to 20, and avoid having to play with the "bad spec" you want to get out of in the first place.
The answer by Raven Dreamer is incorrect in multiple ways. First off, you can respec but it requires a rare multiplayer item drop. You can get it from the purchasable packs, though I am not sure if it's available in all 3 or not.
Also, leveling up to 20 is only done once per class. If you choose another race (or male/female human) you start out at whatever level your other race for that class was. For example, if I level an engineer up to 15 and then unlock the Quarian Engineer it will be 15. If I then level the quarian up to 18 then my human will be 18 as well. One experience bar for all characters in that class.