There’s rarely an issue with Mounted Combat because let’s face it, how often are we able to enjoy the luxury of being mounted when trouble tries to take on an adventuring party?

First the Rules As Written (RAW):

What is or can be classified as a mount:

A willing creature that is at least one size larger than you and that has an appropriate anatomy can serve as a mount, using the following rules [ in the PHb page 198].

So the halfling riding the dwarf doesn’t count (appropriate anatomy), but riding a mastiff does count as a mount for the halfling.

What about getting on and off your mount:

Mounting and Dismounting (PHB198)
Once during your move, you can mount a creature that is within 5 feet of you or dismount. Doing so costs an amount of movement equal to half your speed. For example, if your speed is 30 feet, you must spend 15 feet of movement to mount a horse. Therefore, you can’t mount it if you don’t have 15 feet of movement left or if your speed is 0.

If an effect moves your mount against its will while you’re on it, you must succeed on a DC 10 Dexterity saving throw or fall off the mount, landing prone in a space within 5 feet of it. If you’re knocked prone while mounted, you must make the same saving throw.

If your mount is knocked prone, you can use your reaction to dismount it as it falls and land on your feet. Otherwise, you are dismounted and fall prone in a space within 5 feet it.

There are three main points here:

1. It costs half your movement to mount and dismount. So if you’ve been knocked prone, it will take half your movement to stand up, and the other half of your movement to get onto your mount (if you’re already adjacent to it).
2. If something moves your mount, say someone Shoves your horse (Shield Master feat PHB 170) or the effects of Thunderwave (or similar spell), you then need to make a DC 10 Dex save or be knocked prone.
3. Your mount is knocked prone (Shield Master perhaps), you can then use your reaction to dismount, or if you’ve already used your reaction, you’re now knocked prone next to your prone mount.

In case your mount fly’s and has been knocked prone:

Flying Movement PHB 191
Flying creatures enjoy many benefits of mobility, but they must also deal with the danger of falling. If a flying creature is knocked prone, has its speed reduced to 0, or is otherwise deprived of the ability to move, the creature falls, unless it has the ability to hover or it is being held aloft by magic, such as by the fly spell.

And if your flying mount can hover, it’s till still knocked prone, meaning you’re now dismounted at best and knocked prone yourself. And not on your mount under both circumstances.

Controlling a Mount PHB 198

While you’re mounted, you have two options. You can either control the mount or allow it to act independently. Intelligent creatures, such as dragons, act independently.

You can control a mount only if it has been trained to accept a rider. Domesticated horses, donkeys, and similar creatures are assumed to have such training. The initiative of a controlled mount changes to match yours when you mount it. It moves as you direct it, and it has only three action options: Dash, Disengage, and Dodge. A controlled mount can move and act even on the turn that you mount it.

An independent mount retains its place in the initiative order. Bearing a rider puts no restrictions on the actions the mount can take, and it moves and acts as it wishes. It might flee from combat, rush to attack and devour a badly injured foe, or otherwise act against your wishes.

In either case, if the mount provokes an opportunity attack while you’re on it, the attacker can target you or the mount.

Controlled Mounts

1. Acts on your initiative
2. Can ONLY Dash, Disengage, and Dodge as an action.

Jeremy Crawford had some additional information that he Tweeted:

Disengage: If your mount uses Disengage and then moves you, you don’t provoke opportunity attacks during its movement.

Dodge: A mount dodging doesn’t mean the rider is dodging.

Rider Actions: A controlled mount doesn’t use the rider’s action.

In other words, you don’t need your action to control your mount. You just do it!

Independent Mounts

1. The mount has own initiative
2. Rider is just along for the ride, and the mount may move away from the fight, or it may even attack.

Spells: Find Steed & Find Greater Steed

… unusually intelligent, strong, and loyal steed, creating a long-lasting bond with it.

Your steed serves you as a mount, both in Combat and out, and you have an instinctive bond with it that allows you to fight as a seamless unit.

While your steed is within 1 mile of you, you can communicate with it telepathically.

As to the mechanics:

1. The mount has its own initiative (determined at the start of the combat irrespective of if it is being ridden or not). [most DM’s have them go in the same initiative as PC’s for ease].
2. The mount has its own movement, action, bonus action and reaction.
3. The mount is “unusually intelligent”: without direction it will use its movement etc. in an “unusually intelligent” way. With direction it will probably do what you telepathically say but it is an independent creature and, at the DM’s behest, may do something different subject to its “unusual loyalty“.
4. When you mount your mount you choose if you will control it or if, as it is “unusually intelligent”, you will allow it to act independently as per the rules for mounted combat. It all cases that I can think of, allowing it to be independent is the superior option.

Source: rpg.stackexchange.com

Animal Handling. PHB 178
When there is any question whether you can calm down a domesticated animal, keep a mount from getting spooked, or intuit an animal’s intentions, the DM might call for a Wisdom (Animal Handling) check. You also make a Wisdom (Animal Handling) check to control your mount when you attempt a risky maneuver.

The DM sets Animal Handling checks to control mounts for risky maneuvers, or if the creature is spooked (horses around fire for example).

Vehicle Proficiency. PHB 155
If you have proficiency with a certain kind of vehicle (land or water), you can add your proficiency bonus to any check you make to control that kind of vehicle in difficult circumstances.

Add your proficiency to your Animal Handling check, if you’re not proficient already.

From Xandathar’s Guide to Everything P.82 has the following information for Vehicle Proficiency

LAND AND WATER VEHICLES XGtE 82
Proficiency with land vehicles covers a wide range of options, from chariots and howdahs to wagons and carts. Proficiency with water vehicles covers anything that navigates waterways. Proficiency with vehicles grants the knowledge needed to handle vehicles of that type, along with knowledge of how to repair and maintain them.

In addition, a character proficient with water vehicles is knowledgeable about anything a professional sailor would be familiar with, such as information about the sea and islands, tying knots, and assessing weather and sea conditions.

Arcana. When you study a magic vehicle, this tool proficiency aids you in uncovering lore or determining how the vehicle operates.

Investigation, Perception. When you inspect a vehicle for clues or hidden inform ation, your proficiency aids you in noticing things that others might miss.

Vehicle Handling. When piloting a vehicle, you can apply your proficiency bonus to the vehicle’s AC and saving throws.

Doesn’t really add a lot of information but can give DM’s an idea of DC’s for difficult situations well mounted.

Feat: Mounted Combatant PHB168
You are a dangerous foe to face while mounted. While you are mounted and aren’t incapacitated, you gain the following benefits:
• You have advantage on melee attack rolls against any unmounted creature that is smaller than your mount.
• You can force an attack targeted at your mount to target you instead.
• If your mount is subjected to an effect that allows it to make a Dexterity saving throw to take only half damage, it instead takes no damage if it succeeds on the saving throw, and only half damage if it fails.

Here’s Jeremy Crawford (@jeremyecrawford) – talking about Mounted Combat on a Critical Role PodCast. ARTICLE: JOE MANGANIELLO ON ARKHAN & CRITICAL ROLE

• DM’s should think about how mounts work within the story
• A typical riding mount is more likely to freak-out at combat
• A mount has to be one size larger than the rider, be willing to be mounted, and an appropriate anatomy capable of it (Quadroped, huge or worm-like)
• Controlled vs. Independent actions.
• Int. creatures (ex. Dragon, anything with its own personality) act Independently, which means the mount decides what to do and can take its own turn as normal (attack, dash, etc.). Has its own Initiative
• Controlled: uses PC’s Init. Can only Dash, Disengage, or Dodge
• When your mount provokes an attack of opportunity, the attacker can choose to attack either the mount or the mounted person.

Feats and Mounted Combat

Alert: works just as well mounted.

Athlete: handy if knocked prone off your mount, as you then don’t use all your movements getting up (half movement) and re-mounting (half movement). An argument could be made that Mounting is Climbing your mount so it doesn’t cost half your movement, but there’s also getting situated on the mount and maintaining control of the mount.

Charger: This does not work with mounted combat. It requires you to use your own action (not your mount’s) to Dash.

Defensive Duelist: An excellent defensive option if you don’t use your Reaction frequently, this can be especially useful if you have Mounted Combatant because you can still use it when you take attacks originally intended for your mount.

Dual Wielder: works just as well mounted.

Heavy Armor Master: Just as decent as Defensive Duelist, as long as you’re mounted.

Lucky: When isn’t this feat useful!! I keep reading about home games that ban this feat.

Martial Adept: Battle Master Maneuvers

Mobile: Doesn’t add anything well mounted.

Mounted Combatant: Helps with combat but doesn’t lessen you’re ability to get mounted. Mostly about protecting your mount, which is good since most mounts will be horses and they have rather few HP.

Sentinel: works just as well mounted.

Shield Master: Rather handy with a Lance, being able to push an adjacent opponent away so you don’t have disadvantage attacking. Not as great as pushing an opponent prone and having advantage however.

Resources:

DnD 5e – Practical Guide to Mounted Combat