This is a bit of a thorny question. Might get some details wrong cause it's been a long day, so I'll try to double-check tomorrow for any slip-ups, but:

In Shakespeare's time, plays were a popular entertainment, but they also were frowned upon by the upper class. For most of his career, Shakespeare belonged to 'The King's Men,' and, before that, 'Lord Chamberlain's Men,' two companies that were funded by the nobility and, therefore, given a pass on things. Many acting companies were treated on the same level as vagabonds and rogues, meaning they were not particularly respected. Actors without papers could be arrested, for instance. There was a belief they spread both crime and disease (Fraser/Wells).

At the same time, however, plays were a very popular form of entertainment for the masses. Many venues for their plays offered different "levels" of access: for a cheap price, affordable to the common people, one could enter and watch from the ground. Shakespeare makes reference to these people in several plays, most famously referencing the "groundlings" in Hamlet, which might be a reference to the people who could only afford to enter on the ground level (Bevington). For varying prices, attendees could sit in upraised boxes (Wells), like at a modern football game or concert, or even on the stage itself, a bit like being court-side at a basketball game.

So, to an extent: they were popular with the lower-classes, but aroused a bit more contempt among upper-classes.

As far as the content of the plays: while he did poke fun and get away with jabs at the royalty and nobility many would not, I wouldn't say that Shakespeare qualified as "controversial." Plays had to be reviewed before being put on, and people were aware of the consequences of taking too critical an approach to the royal family (Fraser/Wells). While Shakespeare might have been risqué at times, there's a reason he became a member of a troupe chosen by King James to regularly perform at his court, and it wasn't by being on bad terms with the crown.

Sources:

Russsell Fraser: Shakespeare: A Life Volumes I & II

David Bevington: Collected Works of Shakespeare

Stanley Wells: Shakespeare & Co.