One problem related to brake light on has been reported for the 2013 Toyota Corolla. The most recently reported issues are listed below. Please also check out the statistics and reliability analysis of the 2013 Toyota Corolla based on all problems reported for the 2013 Corolla.
My daughter was driving south on interstate 95 (MD) in the far left lane (fast lane) when her 2013 Toyota Corolla went out of control and went into a 180 degree clockwise spin across 3 lanes of traffic into the far right right lane (slow lane) and when she pumped her brakes the car went into a 180 degree counter clockwise spin back across 4 lanes of traffic almost getting hit by a Jeep before stopping on the left shoulder facing oncoming traffic. Incident happened in rush hour traffic at 9:20am on wet roads. Yellow abs light and traction control light was on, but not the red brake light. Was told by Toyota that the brake actuator assembly failed. In researching this issue, I found Toyota to have a history of brake actuator assembly problems resulting in warranty extensions, service campaigns & recalls on 2007-to 2011 camry's ,2010 prius, 2012 & 2013 tacoma pick ups, Toyota rav 4's.