No you can't hide the spines. The only way to hide them that I know of is if you are a /Dark Armor and you take Cloak of Darkness that will hide your spines and your costume.
"All right, they're on our left, they're on our right, they're in front of us, they're behind us...they can't get away this time"- Chesty Puller US Marine Corps
You cannot hide the spines, but there may be an option that will give you smaller spines, that might not be as obvious. I haven't really looked for sure, but there are metal spines, and "slate" rock spines, and it's possible one of those options would be smaller than the banana shaped regular spines.
You could also make them a color that blends better with your costume or the effects of your Secondary, as the last poster suggested with Dark. Only, instead of becoming transparent, you might make your spines black to blend in with the Dark auras. Taking Stealth from the power pool might also make them less noticable, if you aren't Dark but still have a very intense aura. (You could even give yourself an aura at the Tailor, after 30)
If you don't like the spines at all, you might try Claws, it has some of the same capabilities, although it's not nearly as AoE-centric. There are options to make the claws shorter, though (villain claws are shorter than hero ones) and you can make them black or blend them in with your costume. (Monster Claw hands with brown villain claws tends to make the claws much less noticable)
Hm... I wonder if now that Spines is customizable, there will be an option for Talsorian, or "energy" spines? 
Im playing my first scrapper and its spines. I like the toon but I hate all the spines sticking out of him.
Is it possible to not have the spines show? its bugging me lol.
I dont mean to flame the look, im just asking if i can.
Global: @Fire Beam