In the EU, Regulation (EC) No 852/2004 requires food business operators to put in place procedures based on HACCP principles: identify hazards, set critical control points and limits, monitor them, take corrective action and keep records that prove it. In the UK the same principles apply, and many small kitchens use the Food Standards Agency's Safer Food, Better Business pack. In the US, the FDA Food Code sets the model that states adopt.
The principles are rarely the problem. The records are. A fridge temperature written at the end of a shift is a guess, a cooling log with no time is not evidence, and a missing delivery check is only discovered when an inspector asks. HACCP compliance software is useful when it makes the record at the moment of the check, and makes the gaps visible the same day.