At any Quest Board (or by talking to The Handler) you'll be presented with four options: Assigned, Optional, Investigations and Events. Anything that's listed in the "Investigations" menu count as... well, investigations. You have a chance to unlock investigations by examining monster tracks (during assignments, optional quests, investigations, events or expeditions - as long as you're gaining Research Points, it doesn't matter what the venue is) or breaking monster parts during combat, although you may not have access to investigations until after you complete the quest The Great Jagras Hunt.
Simply check back at the Resource Center in Astera after each mission (basically every time you return to Astera, morseo after hunts or expeditions where you gain a significant number of Research Points from monsters) to see what - if any - new investigations have unlocked and register up to fifty investigations, which can then be undertaken via the aforementioned "Investigations" menu at a Quest Board or by talking to The Handler. Complete any fifty investigations - including as many repeat investigations as you wish - to earn this trophy/achievement.
The Franchise Hunter
Complete 50 investigations.