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Consider this scenario: a College of Swords bard uses dissonant whispers on a creature within 5 feet of him while he is holding a melee weapon. The creature fails the save and has to immediately use its reaction to move away. In my understanding, this movement provokes opportunity attacks. So:

1. Would the caster also be entitled to an opportunity attack?


2. If so, would he be able to augment his opportunity attack with a
   blade flourish?

Two questions here really, but:

1) Yes.

Even though opportunity attacks do not occur for forced movement, the rule for this actually says:

You do not provoke an opportunity attack when you teleport or when
something moves you without using your movement, action or reaction.

Dissonant Whispers however is a bit of an oddball, as it forces the target to: "use its reaction, if available, to move".

So the target is using its own reaction to move and is thus eligible for opportunity attacks.

2) No

You use a blade flourish on your turn as part of the Attack action. An opportunity provides you with an attack (lower-case 'a'), but it is not your Attack action (upper-case 'A').

1. Yes, the caster gets an Opportunity Attack

Since the creature is using its reaction to move away, that movement provokes an Opportunity Attack from an creature whose reach it leaves.

You do not provoke an opportunity attack when you teleport or when
something moves you without using your movement, action or reaction.

In case there was any doubt, Crawford has unofficially supported this ruling in a tweet.

If it leaves the caster's then they provoke from them as well and they can use a reaction to take the attack against them.

You can see Can I trigger an Opportunity Attack by forcing an opponent to move out of my reach? for more examples and discussion about forced movement provoking OAs.

2. No, you cannot use Blade flourish on the Opportunity Attack

Blade Flourish (XGtE, p. 15) only works when you take the Attack action:

Whenever you take the Attack action on your turn [...]

The bard did not take the Attack action this round, they took the Cast a Spell action instead. An opportunity attack is an attack, but it is made outside of an Attack action, so it will not trigger the Blade Flourish ability.

See What does upper-case-A-Attack action vs. lower-case-a-attack mean? for more information on the distinction between the two.

Two parter...

1. Yes, you get an Opportunity Attack

Assuming you have not already used your Reaction in this round. Normally, you do not when a creature is forced to move (via shove, thunderer wave, force wave, etx.), but Dissonant Whispers is different in that the movement is by the creature's Reaction; mandatory, but under the creature's control.

2. No, you didn't use an Attack action for your turn.

Per the feature, "Whenever you take the Attack action on your turn..." You did not take the Attack (big A) action this turn, you took the "Cast a spell" action this turn so Blade Flourish doesn't come into play.