Since there are a lot of topics popping up regarding drop rates and how the Catalogs stack, I thought I'd start this topic just to answer everyone's questions. A lot of this information is taken from the official guide, but some of it is also personal experience.
[Drops]
All monsters ('mobs' from here-on for brevity) have their own drop table. In the table is always a rare drop and a common drop. However, the chance to drop their 'common' and their 'rare' drop differs immensely from mob to mob.
For example, the Adamantoise has a 25% base chance to drop a Platinum Ingot (its common drop) and a 1% base chance to drop a Trapezohedron (the 'Traps' that people typically refer to on the board. For the record, these are the items required to upgrade everybody's Tier 2 weapons to Tier 3 and, thus, their 'ultimate' weapons.)
On the other hand, the Mission 7 mark has a 5% base chance to drop a Tetratic Crown (its common drop) and a 1% base chance to drop a Tetratic Tiara (its rare drop.)
All mobs, however, have the same base chance to drop Shrouds, which is (personal estimation) 1%.
The way the game decides which drop you get is pretty simple. It will first roll for the rare drop. If the roll fails, it will roll for the common drop. In the case of either receiving a common or a rare drop, the game will also roll for a shroud with its chance to drop altered by the Survivalist's Catalog. As was verified by multiple people in the last thread, the chances for a Shroud dropping and a Common/Rare dropping are independent of each other.
[Star Rating]
Your star rating is, as you probably already know, key to getting certain drops. What you might not know is how exactly it affects your chances. Your default star rating for meeting the Target Time is 3 Stars. Star Rating affects two types of drops directly and one type semi-directly. The higher your star rating, the more likely you will get a Rare Drop, but the less likely you will get a Shroud. Due to how the game calculates drops, it also means you are slightly less likely to get a Common Drop as well due to your increased chance of getting a rare.
Here's a simple chart. If the multiplier is 'x0' that means it is impossible to get that drop. Note that it will be in the following format:
Star Rating | Rare Drop Chance Multiplier | Shroud Drop Chance Multiplier
0-Stars | x0 | x8
1-Star | x0 | x4
2-Stars | x0 | x2
3-Stars | x1 (Base) | x1 (Base)
4-Stars | x3 | x1
5-Stars | x5 | x1
Catalogs]
(IMPORTANT NOTE: Best Chance, the Japanese version's equivalent to the Connoisseur's Catalog, had a 1.1x modifier. I AM FAIRLY CERTAIN that this was altered to 1.2x for the US version, but I am possibly wrong. All of this information assumes the 1.2x number.)
Catalogs cannot stack with themselves, but they stack with each other. That means 2x Collector's Catalogs will not increase your drop rate any more than 1, but having one of each WILL increase your drop rate for each type of item. Note that it also doesn't matter which character has the Catalog(s) equipped, as long as they're in your active party.
All Catalogs apply their bonus BEFORE the post-battle Star Rating bonus. This is important when calculating drop rates.
There are three types of Catalogs: Survivalist's, Collector's and Connoisseur's.
Survivalist's Catalogs increase your chance to obtain a Shroud. It increases the base chance by 1.5x. This means, at a 0-Star rating, assuming the base drop rate for a shroud is 1%, you have a 12% chance of getting a Shroud.
It's actually quite easy to game the system by equipping a Survivalist Catalog and getting into a fight with multiple very weak mobs and waiting roughly a minute before killing them in order to get a 0-Star rating. It's a pretty effective way to farm Shrouds if you need them.
Collector's Catalogs increase your chance to obtain a mob's common drop if the game tries to roll for one. You will get a chance at a common roll if you do not get a rare drop first from that mob. It increases the base chance of the roll by 1.5x, like the Survivalist's Catalog.
Connoisseur's Catalogs increase your chance to obtain a mob's rare drop if the game tries to roll for one. You can only get rare drops at 3-Star ratings or higher, and the game will always roll for the rare first. It increases the base chance by 1.2x.
As a result, you probably tell now that if you're looking for rare drops, you'll want to strive for the highest Star Rating you can and keep a Connoisseur's Catalog on at all times. If you're looking for common drops, you want to try to game the system so that you get a 2-Star Rating or worse and keep a Collector's Catalog on. If you want to solely farm shrouds, then you will want to make sure you're using a Survivalist Catalog and all of your star ratings are 0-Stars as noted before.
If you're looking for getting SOMETHING out of a battle, no matter what it is, then keep all three on and get a high star rating!
[Drop Rates]
For reference, here are some drop rates of mobs you will commonly be farming. Note that all of these rates assume that you are getting a 5-Star post-battle rating. The base chance (without calculating in Star ratings) is also included in parentheses for reference.
-=Adamantoise=-
(1% Base Chance to drop a Trapezohedron)
(25% Base Chance to drop a Plat. Ingot)
5% Chance to drop a Trapezohedron w/o Conn. Cat.
6% Chance to drop a Trapezohedron w/ Conn. Cat.
25% Chance to drop a Plat. Ingot w/o Coll. Cat.
37.5% Chance to drop a Plat. Ingot w/ Coll. Cat.
-=Long Gui=-
(5% Base Chance to drop a Trapezohedron)
(25% Base Chance to drop a Plat. Ingot)
25% Chance to Drop a Trapezohedron w/o Conn. Cat.
30% Chance to Drop a Trapezohedron w/ Conn. Cat.
25% Chance to drop a Plat. Ingot w/o Coll. Cat.
37.5% Chance to drop a Plat. Ingot w/ Coll. Cat.
-=Shaolong Gui=-
(5% Base Chance to drop a Dark Matter)
(25% Base Chance to drop a Gold Nugg.)
25% Chance to Drop a Dark Matter w/o Conn. Cat.
30% Chance to Drop a Dark Matter w/ Conn. Cat.
25% Chance to drop a Gold Nugg. w/o Coll. Cat.
37.5% Chance to drop a Gold Nugg. w/ Coll. Cat.
-=Sacrifice=-
(5% Base Chance to drop a Scarletite)
(25% Base Chance to drop a Perfume)
25% Chance to Drop a Scarletite w/o Conn. Cat.
30% Chance to Drop a Scarletite w/ Conn. Cat.
25% Chance to drop a Perfume w/o Coll. Cat.
37.5% Chance to drop a Perfume w/ Coll. Cat.
