Do we know what the Big Three (Asimov, Clarke, and Heinlein) thought of each other's writing? Did they have any favorite works of each other?
Asimov vs Heinlein
Asimov and Heinlein did have some disagreements, according to this article on io9:
Primarily their conflict became a political disagreement, as Asimov revealed in his posthumous 1994 autobiography.
and later on:
Living longer than Heinlein allowed Asimov to have the last word in the debate, bashing the release of Heinlein letters Grumbles from the Grave.
However, it is also mentioned in that article that Asimov's favorite Heinlein novel was Double Star.
As for Clarke, the relationship to both Asimov and Heinlein is expressed in the "The Big Three" paragraph on Arthur C. Clarke's wikipedia page.
Clarke vs Heinlein
Clarke and Heinlein began writing to each other after The Exploration of Space was published in 1951, and first met in person the following year. They remained on cordial terms for many years, including visits in the United States and Sri Lanka. In 1984, Clarke testified before Congress against the Strategic Defense Initiative (SDI). Later, at the home of Larry Niven in California, Heinlein attacked Clarke verbally over his views on United States foreign and space policy (especially the SDI). Although the two reconciled formally, they remained distant until Heinlein's death in 1988.
Clarke vs Asimov
Clarke and Asimov first met in New York City in 1953, and they traded friendly insults and gibes for decades. They established a verbal agreement, the "Clarke–Asimov Treaty", that when asked who was best, the two would say Clarke was the best science fiction writer and Asimov was the best science writer. In 1972, Clarke put the "treaty" on paper in his dedication to Report on Planet Three and Other Speculations.