Build is a rogue assassin/hexblade(5)/bard(2) (or bard 1/sorcerer 1) , has elven accuracy, is dual wielding a shortsword (vex) and scimitar (nick, scimitar is the pact weapon), and has thirsting blade and eldritch smite as the warlock 5 invocations, so if the character goes before his target in round 1 he has advantage on all 3 attacks, his bonus action will be using hexblade's curse so will rolling a total of 9 d20's for 3 attacks and critting on a 19 or 20, and burning a 3rd level pact magic to eldritch smite the first 2 times he crits a significant opponent.   If i.can keep advantage I'll get the 9 d20 rolls each round making at least 1 crit a round fairly likely.   If I don't have advantage, I could attack with short sword (vex) first, if I hit it'll grant advantage on the next attack, otherwise burn a lucky, then sneak attack, use a d6 for trip/prone and the third attack has advantage.  Which brings me down 7 d20 rolls a round.  Once i have bard 1 Ill be able to Faerie fire twice  a day (or use silvery barbs, to make an opponent fail a save and grant myself an advantage on my next attack) but if I was able to keep the opponent from getting up, I'll get the 9 rolls a round without spending resources.

So how could I keep a prone opponent from getting up (apart from reducing him to zero hp or grappling him)?

Anything that applies the Restrained condition is your friend here, since a Restrained creature has a speed of 0, and you can't stand up if your speed is 0.

Become proficient in improvised weapons, and use a net as a weapon for one of the weapons. And have a steady supply of nets.

Grapple.

Any effect that make your speed 0 will do, such as the Grappled, Paralyzed, Petrified, Restrained or Unconscious condition.

Or, you could try sitting down on top of the Prone opponent.

Quote from GeijerFeyLock >>

Or, you could try sitting down on top of the Prone opponent.

Most DMs would probably handle that as a Grapple.

Can you get access to Ensnaring Strike without multiclassing?

Quote from SmiteMakesRight_3_5 >>

Can you get access to Ensnaring Strike without multiclassing?

 Now that the Bard Magical Secret no longer get choose it, I believe it's now only accessible to Rangers and Paladin Oath of the Ancient. 

Quote from Plaguescarred >>

Quote from SmiteMakesRight_3_5 >>

Can you get access to Ensnaring Strike without multiclassing?

 Now that the Bard Magical Secret no longer get choose it, I believe it's now only accessible to Rangers and Paladin Oath of the Ancient. 

There is the Azorius Functionary background in Guildmaster's Guide to Ravnica, but I was hoping something more wildly accepted existed.

I was thinking in new core rulebooks for 5E14 3PP it's probably the only source that does but i'd have to check if there's any other. 

Quote from wagnarokkr >>

Quote from GeijerFeyLock >>

Or, you could try sitting down on top of the Prone opponent.

Most DMs would probably handle that as a Grapple.

 Or, in the case of a Goliath character sitting on a halfling, it would be more of a seat cushion squash.

Place an immovable rod on their chest and click the button? Of course they could always eventually figure out how it works and just turn it off, but that might take a while.

Agent of order from Planescape is a personal favorite. If your DM eschews the prerequisite of needing the background feat, or lets you use the given background feat of Scion of the Outer Planes (Law) instead of the usual ones granted.

It allows you to add a D8 damage die and triggers a WIS save or become restrained proficiency times per long rest. It also increases any stat by 1, so it is a cool half feat that adds some extra flavor and utility to your combats.