I'm very curious to see how Bungie has chosen to develop the shooting mechanics of the game. It has been confirmed that "ADS" will be featured but beyond that, there isn't much to go on. I'm going to talk(probably a lot and in a very confusing manner) about what I think the mechanics will be but I'd really like to get the community's opinion!

I'd first like to define a few things to create a foundation for discussion and to add some clarity to the conversation. Use the things I've outlined below to discus what you think will/wont appear.

The hardware:

ADS: My brief tour of wiki didn't lead me to any formal or universally accepted treatments of FPS mechanics but I believe I have a pretty good grasp on the gist: The player can increase accuracy(visually, through decreased recoil, etc) by utilizing the weapons sights.

Some games, a la Ghost Recon:Future Soldier, have three states of aiming:
(1) No aim. No or uncontrolled target reticule. High recoil.
(2) Half-aim/weapon raised. Increased accuracy, decreased recoil. Limited field of vision. Useful target reticule.
(3) Aim-Down-Sights. Iron-sights or attached optics consume a large portion of the player's field of vision. Highest accuracy, lowest recoil. Often includes a "scope in" or "scope out" time delay. Reduced movement speed. May include "breathing" effects.

Arcade-Style: If I'm not mistaken (please don't hate me), this is the term for the style of play featured in the Halo franchise. Weapons with out optics do not feature an ability to aim. Weapons with optics can aim, usually with perfect accuracy. "Sniper" weapons may feature a breathing/steadying mechanic. The amount one shoots, depending on the weapon, will affect the fixed target reticule which itself may vary in size to indicate increased recoil and decreased hit percentage or accuracy.

Movement and Environment:

Every shooter chooses the extent to which its players are able to manipulate their avatars through the environment(although none of these choices are made under the same circumstances that concern Destiny. An important question to be revisited later). I've created the following list to rank one's fluidity in a game.

1: Walking. Running. Jump. Crouch. (Standard)
2: Standard. Prone. Crouch(stationary). Crouch(movement). Variable walk/run speeds. Diving/sliding/crouch-running/cover-sliding. Specific, often contextual, climbing, dropping, etc.
3: Standard. Most or all of category 2. General ability to climb or otherwise fluidly interact with the environment. May include dynamic environments.
Other: Ussain Bolt-mobile. Often a specific and central feature of the game.

The ability of a character to move is very much attached to the game's cover system. Using the above framework, example games include:
1: Halo
(1.5: Call of Duty)
2: Battlefield 3
(2.5: Killzone 4, Ghost Recon:Future Solider, Crysis 2)
3: Metal Gear Solid 4
Other: Vanquish, Dark Void, Watch_Dogs

Last but not least, there is level design. Based upon the vision for the game and the combination of the previous factors, level design will dictate gameplay. Dur. How will this factor into an MMOish game? I have no idea but I'd like to. Obviously the world of Destiny is open enough for vehicles which allows for roaming or patrolling enemies such as wildlife and mobs(Borderlands). The screen shots of the alleged space zombie environments also appear to be very open*, providing for a classic MMO dungeon experience where not every enemy must be engaged.

* http://www.destinyhub.net/content/46-A-scene-by-scene-analysis-of-the-PlayStation-event

Personally
, I'm a fan of creating environments that make sense with the story/atmosphere; I'm not a fan of my characters being unable to lay down. I'm often removed from the experience of the game when I wonder why these elite soldiers are running around open environments without the faintest excuse for cover; or why commanders only seem to choose fields, mezzanines, large highways and hangar bays as battle grounds. I'm also not a fan of covering 100% of the map in chest-high walls (or empty rooms with sophisticated mechanical systems which exist solely to deploy chest-high walls). I think Mass Effect has a good mix. while it is very explicitly a "cover-based" shooter, I think the genre should be regarded simply as "shooters" while anything else gets renamed "Full-on Bronson"...but I digress...

It would make me a little sad to see them stick too close to Halo with this new IP. The breadth of environments allows for a nearly endless variety of maps and zones, each of which are able to present wholly unique challenges for the players. It would be a huge loss (IMO...IMO...) to have the different maps be environmental backdrops for the exact same gameplay, especially if its simplistic.