Is there any way to get an instant or sorcery on the field as a permanent? What would happen if this occurred?
There's a clear boundary between the two types of cards (one having a 'lasting' effect, the other a 'momentary' effect), and as far as I can see that's intentional. In the history of Magic: The Gathering, a handful of instant/sorcery cards with transform and/or manifest have been printed, and with some other cards it would be possible to make permanents of them, except for some very specific rules (mentioned by @doppelgreener) preventing this:

304.4. Instants can’t enter the battlefield. If an instant would enter the battlefield, it remains in its previous zone instead.
307.4. Sorceries can’t enter the battlefield. If a sorcery would enter the battlefield, it remains in its previous zone instead.
701.27d If a spell or ability instructs a player to transform a permanent, and the face that permanent would transform into is represented by an instant or sorcery card face, nothing happens.
701.33f If a manifested permanent that’s represented by an instant or sorcery card would turn face up, its controller reveals it and leaves it face down. Abilities that trigger whenever a permanent is turned face up won’t trigger.

Even if those rules didn't exist, the text on an instant or sorcery indicates what happens when the spell resolves. So if you could somehow get a Lightning Bolt on the battlefield as a permanent, it won't do any damage, not even when it enters the battlefield.