Before Disney, the 501st Clone Battalion can be said to be Palpatine's personal favourite, if he ever harboured any bias for particular clones.
The 501st distinguished themselves as the best performing battalion in the First Battle of Geonosis. As a result, Palpatine had them brought to Coruscant for additional secret training. They are subsequently deployed as Anakin's personal battalion, while certain units of the battalion are secretly attached, sometimes on a more permanent basis, to other units to help accomplish secret missions for Palpatine himself, among which include securing vital tech and equipment necessary to the eventual construction of the Death Star. On the eve of the execution of Order 66, all elements of the 501st are secretly transferred back to Coruscant again, this time commanded by the newly christened Darth Vader to destroy the Jedi Temple. During the Empire, the reorganised 501st Stormtrooper Legion remains under Vader's command, and comes to be known as Vader's Fist.
Under Disney, however, much of the above cannot be considered canon since most of it is sourced from Battlefront II, which is considered Legends. From the Clone Wars series, I know that the 501st remains Anakin's personal unit, and is certainly the one that followed him into the Jedi Temple. Is there anything under Disney's canon that I do not know of? Is the 501st still purpose-trained for the specific mission of storming the Jedi Temple? What about thereafter? Did they enjoy any special status under the Empire?