Pg. 234 PHB, highlights are mine..
You whisper a discordant melody that only one creature of your choice within range can hear, wracking it with terrible pain. The target must make a Wisdom saving throw. On a failed save, it takes 3d6 psychic damage and must immediately use its reaction, if available, to move as far as its speed allows away from you. The creature doesn't move into obviously dangerous ground, such as a fire or a pit. On a successful save, the target takes half as much damage and doesn't have to move away. A deafened creature automatically succeeds on the save.
Assume my character is grappled and has successfully cast Dissonant Whispers. Also assume the creature failed its saving throw and has a reaction available.
Does the creature release the grapple in this scenario?
I found this text on pg. 234 PHB which explains the rules for the Grappled condition:
… The [grappled] condition also ends if an effect removes the grappled creature from the reach of the grappler or grappling effect, such as when a creature is hurled away by the thunderwave spell.
Also, on pg. 195:
Moving a grappled creature. When you move, you can carry or drag the grappled creature with you, but your speed is halved…
Best Answer
Yes
The text of the spell says that the creature must move away from the caster. If the grappler carries or drags the caster with it, there is no change in distance between the grappler and the caster.
The only scenario where I can imagine where the caster remains grappled is if the target has a speed smaller than its reach; e.g. it has a speed of 5 ft but a reach of 10 ft.
I don't know of any other RAW that would contradict this. However, I think that the spirit of the spell is that the target recoils from the caster, and as a GM I would be willing to entertain the idea that failing the save implies that a creature would release the grapple.