Generally, not very big. We have short memories as fans, players chop'n'change too much to establish loyalty and we suffer from large amounts of media driven faux-outrage. State of Origin is as good as rivalry gets.
From time to time, two teams will have a string of games where they're evenly matched, which makes for good ongoing fan banter (Think Melbourne Storm and Manly Sea-Eagles : 2007 - Ongoing).
Us Bulldogs ... we have traditional rivals like Parramatta (evenly matched in 80's), St George Dragons (geographical neigbours) and more recently The Sydney Roosters (due to them winning the comp in 2002 when we were removed from finals contention due to salary cap rule breaches, and us beating them in 2004 Grand Final). You can add Manly Sea Eagles to that list since 2012, when we stole their coach from right under their noses.
Fights occur in crowds from time to time, but are generally frowned upon in the media. In 2003 we had a rather large outbreak of fights with Roosters fans one particular game, and in 2004 we were subject to gang-rape investigation. To this day other fans will use it as an excuse to label the whole of our fanbase as thugs and call anyone in our colours some racist/sexist/misogynist slur. Which (as you can imagine) often results in punches being thrown.