Paul's monologue from A Chorus Line is the first that comes to mind. During the tour a few years ago I think I actually found myself holding my breath, it's so intense and focused, and I still remember the images that came to me as he described his journey. Apparently in the original production the air conditioners were shut off during it, so there would be total silence.
I loved George Rose's opening monologue that began "The Mystery of Edwin Drood". His delivery made you feel that you were in for a very enjoyable evening of theatre.
This is a fascinating thread. I had never thought about the question of monologues in musicals before. I watched the Richard Burton video and was mesmerized. However, I found myself wondering whether his monologue was meant to be sung instead of spoken, given that it was spoken to music. Also, I wonder whether the reason there are so few monologues in musicals is that a character's rumination usually takes place in song, even if the piece would be equally (or more) moving as a spoken monologue. The examples I can think of are "It's a Puzzlement," from The King and I, "Soliloquy," from Carousel, "I've Grown Accustomed to Her Face," from My Fair Lady, and Jean Valjean's soliloquy (whose name I've forgotten) from Les Miz, where he vows to leave JVJ behind.
Audrey, the Phantom Phanatic, who nonetheless would rather be Jean Valjean, who knew how to make lemonade out of lemons.
This may technically be dialogue, but I've always loved the "butterfly" monologue and Bobby's "motel room" exchange from "Company."
My favorite monologue from a musical, though, is the "Have it Your Way" one from "Assassins." I'm so grateful it was recorded in its entirety on the revival album!
I always thought the two monologues from A Little Night Music during Send in the Clowns always was very powerful.
"Whenever I get gloomy with the state of the world, I think about the arrivals gate at Heathrow Airport. General opinion's starting to make out that we live in a world of hatred and greed, but I don't see that. It seems to me that love is everywhere. Often it's not particularly dignified or newsworthy, but it's always there - fathers and sons, mothers and daughters, husbands and wives, boyfriends, girlfriends, old friends. When the planes hit the Twin Towers, as far as I know none of the phone calls from the people on board were messages of hate or revenge - they were all messages of love. If you look for it, I've got a sneaky feeling you'll find that love actually is all around."