For nearly every launch in the history of Kennedy Space Center, the vehicles have rolled out the east side of the Vehicle Assembly Building to the Launch Complex 39 pads.
Why the east side? Because that's the side that faces towards the launch pads.
But there have been rare exceptions.
The Saturn V stack for Apollo 13 leaves High Bay 2 on August 8, 1969. It was transferred to High Bay 3 on the east side to add the command and service modules. Image source: NASA.
The Saturn V rockets for Apollo 10 and Apollo 13 found themselves in High Bay 2 on the west side as processing for other vehicles on the east side limited available space. Apollo 10 was the only Saturn V to launch from LC-39B. Apollo 13 was forced into HB-2 until space was available on the west side.
STS-106 Atlantis emerges from High Bay 2 on August 12, 2000. Image source: NASA.
Space Shuttles were never forced to the west side due to processing delays, but STS-106 Atlantis was used to demonstrate the viability of a renovated HB-2 for a project called Safe Haven. The idea was to provide a location on the west side for protection from incoming hurricanes. The crawlerway into HB-2 was also restored.