Can I play a Sorcery or Instant without paying its mana cost, in any way?
There are a number of such cards. Recently, Omniscience gave you permission to cast any spell (instant, sorcery, creature, whatever) from your hand without paying its cost.

There are even entire mechanics based on it, such as Cascade, or the cycle of Hideaway lands.

Epic Experiment is the instant/sorcery version of Genesis Wave, and Mind's Desire is its predecessor. Master of Predicaments was just released, and Oracle of Bones is recent. Sunforger is older, but one of the most powerful equipment for a R/W deck: not only is the instant you cast free, but you search your deck for it, rather than needing it in your hand.

Heck, Planeswalker's Mischief lets you cast an instant or sorcery spell for free from your opponent's hand.

Note that when you're casting a spell without paying its mana cost, {X} can only be 0, you must still pay any required additional costs, and if you want the effect of something that comes from paying an optional cost (eg, Kicker), you'll need to pay the optional cost. Effects which look at the mana you paid for the spell (such as the hybrid mana spells which care about the colors spent) will see that no mana was spent at all, and Trinisphere will still force you to pay {3}.