Summary
For purposes of this discussion, "spell" means a spell of 1st level or higher. The short, short version of my decision is that you can only cast a maximum of two times per round, one of which must be a cantrip. This is based on the rule given on page 202 of the 5E Player's Handbook under Casting Time and Bonus Action: If you cast a spell with a bonus action, you can't cast another spell except for a cantrip with a casting time of one action. Now, part of the confusion comes from the issue of "turn" versus "round". A "round" is when all participants in combat have each taken their individual actions or "turns". Some of the rules are based on what happens in the entire "round", which is causing the confusion.
Discussion
Part of my ruling is based on my 1st edition preferences. Casting more than one spell per round seems excessive, but since the rule specifies one must be a cantrip, I can accept that. Another support for my decision is the spell Haste, which gives you an extra action, but that action cannot be used to cast a spell. Spellcasters at higher levels are dangerous enough, allowing a wizard to shoot two Lightning Bolt spells in one round for example would be ridiculous. If you want a roleplaying justification: casting a spell is a mighty mental effort, a cantrip is a quick, simple spell that you can cast on the heel of a major spell, but two major spells in quick succession is too much effort for most mortal spellcasters. Also, the spell words and gestures are delivered in a specific pattern and timing, belting them out faster like an auctioneer while you are under the influence of Haste would cause the spell to fail (But see Sorcerers below).
Reactions
The issue arises when you consider your reaction action during a round and that some spells like Feather Fall, Shield and especially, Counterspell, are cast using your reaction. So, argument goes, you could cast one spell with your bonus action, a cantrip with your action during your turn and another spell with your reaction at some point during the round, allowing you to cast three spells in a round. The rule on pg. 202 doesn't mention reactions, because reactions don't take place during your turn, so it doesn't say you can't, but it also doesn't say you can. However, see Sage Advice below.
For the reasons given above, I'm going to maintain the rule of "One Spell, One Cantrip". If you cast a spell using your reaction before your turn comes up, you can only cast a cantrip on your turn.
Sorcerers
Sorcerers have the ability to use the Quicken Spell Metamagic feat to change a spell's casting time from one action to a bonus action. However, if they do so, they are still limited to using their regular action to cast a cantrip. If they use a reaction to cast a spell, then cannot use Quicken Spell to cast a one action spell as a bonus action spell. Although, in theory, they could use Quicken Spell to cast a cantrip as a bonus action, obeying the "One Spell, One Cantrip per Turn" rule. And leaving their regular action available to make a weapon attack.
Sage Advice
I found a combined Sage Advice article which manages to muddy the issue. Under Sorcerers, it specifies that they cannot use Quicken Spell to cast two 1st level or higher spells in a round, they must follow the "One Spell, One Cantrip per Turn" rule
