When looking through the combat rules I realized under Being Prone it states:
You can drop prone without using any of your speed.
There doesn't seem to be a reference that this needs to be on your turn or during your move or whatever else.
Notably, there are ways to move off of your turn such as dissonant whispers or the Ready action.
Can you voluntarily drop prone when it is not your turn (when no other effect is making you fall prone)?
Short Answer: Just falling to prone counts as movement and still requires to be on your turn, but it's something that can easily be setup as a reaction.
Dropping to prone is found inside the movement phase rules, which means that it is used in your turn's movement:
You can drop prone without using any of your speed.
So consider it an action that is in your movement action, but uses 0 ft. So even if you use your full 30 ft (or however much movement speed your character has), you can still drop to prone at the end, because it costs 0 ft of movement.
Now, I'm glad you brought up the Ready action, because that is exactly what you can do with it.
Sometimes you want to get the jump on a foe or wait for a particular circumstance before you act. To do so, you can take the Ready action on your turn, which lets you act using your reaction before the start of your next turn.
First, you decide what perceivable circumstance will trigger your reaction. Then, you choose the action you will take in response to that trigger, or you choose to move up to your speed in response to it. Examples include "If the cultist steps on the trapdoor, I'll pull the lever that opens it," and "If the goblin steps next to me, I move away."
When the trigger occurs, you can either take your reaction right after the trigger finishes or ignore the trigger. Remember that you can take only one reaction per round.
This means that dropping to prone can either be done on your turn's move action, or during the move action of the reaction from readying an action outside your turn. To do this, you'll give your DM a trigger for your reaction to occur, then go prone. You can even use your movement speed first and then go prone, or go prone and crawl up to half your speed, because dropping to prone uses 0 ft of movement.
And yes, you can use your turn's move, and then move again on your reaction.
There doesn't seem to be a reference that this needs to be on your turn.
PHB 190:
On your turn, you can move a distance up to your speed. You can use as much or as little of your speed as you like on your turn, following the rules here...
Emphasis is mine.
The base assumption is that you take actions on your turn. Just because it doesn't subtract from your possible movement speed doesn't mean it doesn't have to happen on your turn.
However:
There are a number of ways the prone condition might be triggered on you at a different point in initiative order. These include:
Your best bet for dropping prone not on your turn is going to be the use of the Ready Action. However, you still need to set the condition on your turn, so that you can use your reaction later.
PHB 193
Ready action on your turn so that you can act later in the round using your reaction. First, you decide what perceivable circumstance will trigger your reaction. Then, you choose the action you will take in response to that trigger,
On your turn, you can move a distance up to your speed and take one action ...
Various class features, spells, and other abilities let you take an additional action on your turn called a bonus action....
Other Activity on Your Turn
Your turn can include a variety of flourishes that require neither your action nor your move.
You can communicate however you are able, through brief utterances and gestures, as you take your turn...
Notice a theme here? "On your turn... on your turn... your turn can include... as you take your turn..." But then it says this:
A reaction is an instant response to a trigger of some kind, which can occur on your turn or on someone else's.
This is the only mention in the action-timing rules of being able to do anything during someone else's turn. There's no explicit rule that you must use a reaction to act outside your turn, but it's further supported by this line:
If the reaction interrupts another creature's turn, that creature can continue its turn right after the reaction.
Not "if one creature interrupts another creature's turn", but "if the reaction interrupts". The implication is that if you interrupt another creature's turn, you are taking a reaction. There's no provision for a "free action" on someone else's turn. (So-called "free object interactions" are part of your movement or your action, and are described under the heading "Other Activity on Your Turn".)
Conclusion: While dropping prone costs zero feet of movement, you can only do it on your turn (or as a readied action), because you can generally only do anything on your turn.
