I found this[1] talk about LHCb results and future direction illuminating. He explains the trigger setup during the first few minutes, later on he explains how they're searching for new physics.
For run 2 of LHC they used 50000 CPU cores for their software triggers, after the hardware trigger has reduced the 40 MHz input rate down to 1 MHz. The final output of the software triggers is 12.5 kHz, which is persisted to disk. Keep in mind this is just for the LHCb detector.
For run 3, they're planning to remove the hardware trigger bit, running the software triggers directly from the 40 Mhz signal. This would allow them to reprogram the triggers during the run, in case some new interesting theory comes along which for some reason has a signal their current trigger won't identify.