Most of the trophy fish use the same model with different skin hue for different species. The same size model is used for all fish, "weight" is ignored in that regard.
Lobsters and groupers are an exception; currently these are not displayed at all.
Most trophy fish are wielded in the off hand. If you merely wield the trophy fish, it will not show on your character. You have to type "z" to force the wielded items to display. (Exception, the lobster trophy fish do not display.)
Press the "z" key to toggle displaying your equipped items. (With a ranged weapon equipped, you may have three display options, in which case it will step through them.) One of the display options will show the trophy fish being held in your off hand.
Since trophy fish are wieldable objects, they do not stack in your inventory so each fish takes a slot; except the wieldable cooking ingredient fish, which do stack.
There are a few major subcategories of trophy fishes:
Large normal fish
Most (older) trophy fish are a variation of a normal or buffing fish, but unlike their non trophy counterparts, these trophy fish cannot be eaten as fish nor used in recipes. They are available in a partially populated range of sizes, specified in pounds of weight. Since there is a consensus of which few sizes are available, it appears that these are generated off a table, not at random over a range of values.
These trophy fish do nothing, but can be equipped in the off-hand and displayed (with the exception of the trophy groupers and trophy lobsters which can not be displayed.)
While the main purpose of these fish is to be displayed proudly to other players, they can be fed to hunter pets that eat fish, but only producing the same results as feeding a normal fish of that type. There is no benefit from the specialness nor extra size.
These non-stat trophy fish are not bound, and can be sent via mail, sold to vendors, sold on the auction house, and traded. The demand in the auction house is low, but they do sell, and you can get surprisingly much money for your trophy fish on occasion. The item type and subtype for these fish is Miscellaneous, Junk; this is where you find them in the auction house.
These fish include the following:
Large ingredient fish
The Burning Crusade added two fish which can both be trophy fish or can be used in cooking recipes (see above):
These have the Miscellaneous, Junk type and subtype, even though you might expect them to be Trade Goods, Meat items.
Artisan Fishing quest trophy fish
The quest fish can be wielded in the off hand as a trophy fish, but the Feralas Ahi can only be caught when you are on the quest, the fish has the unique property, so you can only have one, and the fish must be turned in to complete the  quest, which formerly granted artisan level fishing skill, raising the fishing skill cap from 225 to 300. Before this, people wanted to turn in the quest to advance their fishing, but now catching the fish as a trophy to wield and leaving the quest open is slightly more common.
Faction trophy fish
Patch 2.3.0 (13-Nov-2007): added two more trophy fish. Unlike previous trophy fish these provide a bonus when wielded.
These bind on pickup and cannot be traded, and are very rare (estimated roughly 0.1% of catches in their respective locations.)
- adds +5 Stamina when wielded off hand. Caught in the waters in Orgrimmar.
- adds +5 Stamina when wielded off hand. Caught in the waters or lava in Ironforge.
These very rare fish provide the unique novelty of a weapon that looks like a fish. These bind on equip, so can be sold at the auction house. Due to their uniqueness, they can fetch a high price. They are weapon items. All three of these weapons are one hand weapons, and none of them are unique. That means that any class can wield them. Single wield characters can only wield them in the main hand. Dual wield characters can wield them in either hand, or if you have two, they can be dual wielded. (See Foodfight for more food weapons.)
Both Old Crafty and Old Ironjaw are achievements for fishing in a particular area.
Attempting to catch Old Crafty will fulfill the Orgrimmar half of - Fish something up in Orgrimmar and Stormwind - when you catch your first catch in Orgrimmar. While you are there, Alliance, you may want to do Ragefire Chasm for the achievement.
For the achievement - Catch one of the rare fish in the list below (Need 1 out of 10):
Seven are trophy fish, heaviest of species (rarest):
Three are very rare weapon fish: