This is mainly referencing those who are getting in to running Advanced Quests, but there are some things that apply to more than just advanced quests.
First up are units that drop. When hovering over the item, check the top-right corner to see how many "slots" it has. My general rule of thumb is if any unit drop has 4 slots, I will at the very least pick it up to see what affixes are attached to it. Some augments on a 4 slot unit will elevate the amount it is worth if you have a personal shop pass, but it is also good to collect for when you are ready to do your own affixing.
For weapons I typically don't pick them up unless they are at least 5 slots, with a few exceptions.
In the listen circumstances below, ALWAYS check these drops.
If you encounter the Apprentice emergency code (purple clothed human battle), pick up the weapons and units that he drops. The Apprentice Soul can be very valuable to either sell of use on your own equipment. You can always select the item in your inventory and "Check market price" based on the augments and number of slots it has.
The ARKs Clone Emergency code (when you encounter what looks like a party of other players) has a chance of containing the augment "Modulator", which on Ship 2 currently is selling for ridiculous amounts even on a 2 slot weapon/unit. So regardless of the rarity of the weapons/units they drop, pick them all up and check their augments. You can land a really nice payday if you get anything above 3 slots. Again, you can also use it for your own equipment if you like.
Extreme quest equipment drops are always worth picking up since you don't actually collect the item until you've completed the mission. When looking at items to choose, the weapons/units that have augments prefixed by Alter are good to collect. The more slots, the more it is worth, and once more you can add it to your own equipment if you like. When running extreme quests, it is also good to use any spare Meseta Boosters you have. Extreme quests drop a lot more meseta than most other quest types.
Finally running SH Advanced Quests? Check every green item container. Why? On certain maps you have the chance to find Photon Arts that currently sell for ridiculous prices at level 10. Million Storm is a good example. It is currently selling between 400K and 500K on Ship 2. I sold several for around a million a piece before they dropped in price. The point is that all PA discs that drop on these maps have a chance at being level 10 and can come from enemies and containers so don't forget to check them all.
That's all I got for now, if anyone else wants to add to the list of drops to always check, feel free to do so.