Shards have the ability to see the future, but even for them it is difficult and imprecise. Certain shards are particularly are better or worse than others, generally relating to their Intents:
Preservation: Very good at seeing the future, to the point of correctly predicting a prophecy being interfered with by another Shard over thousands of years in such a way that the original intent still comes true. Preservation is naturally going to be good at this, as preserving something requires planning for the future.
Ruin: Is one of the worst shards at seeing the future. He cares about destruction, so generally the best he can do with his future-sight is delay destroying something now to destroy more stuff later.
Endowment: Endowment plans for the future by giving individuals gifts and trusting they make the right choices to use them. It's unclear exactly how well Endowment is able to ensure things go to plan, but we have seen at least one example of Endowment managing to get someone into just the right place at just the right time.
Honor: Mediocre at best, it ties into Honor Before Reason—even if you know your actions are going to have a poor effect, you have to do it anyway, because it's right. Plus a large portion of Honor's focus seems to be specifically on honoring Oaths, something which is entirely based on the past and present.
Cultivation: Much better, possibly one of the best Cultivation is all about planting and nurturing things that will come to fruition in the future which only works with excellent knowledge of consequences.
Odium: Better than Honor, as everyone on Roshar insists seeing the future is of Odium, but Cultivation seems to have him beat. True hatred of his enemies involves long-term cruelty and planning, but he claims to be about all Passionate emotions which would imply a recklessness and lack of long term planning, making it hard to say exactly where he would land on the scale.
Why Allomancy shatters the crystals in the Pits of Hathsin: Allomancy is the power of Preservation, and the crystals in the Pits gather the body of Ruin, so the powers are diametrically opposed. Allomancy would naturally disrupt Ruin's power. Preservation probably deliberately set that whole thing up. Which brings up the interesting question of what happens if you burn atium around the crystals.
At first, it looks like Allomancy having a hard time Pushing or Pulling any metals partially or completely inside someone's body is a matter of plot convenience. It would be too easy if Mistborn could rip the metals out of someone's stomach or the spikes in a Inquisitor's head. There's an unstated reason for this, though: the Spiritweb is made of Investiture, which is also what Allomancy is driven by, and Investiture resists other Investiture. This is also why a sufficiently filled metalmind is undetectable to Ironsight, and of course why a sufficiently powerful Allomancer can overcome the limit and still push on metals inside of someone. Hemalurgic spikes rip out pieces of said Spiritweb.
The Ars Arcanum, presented as an in-universe document, notes that Hemulurgy is a Game-Breaker as it (at least in theory) allows anyone to access and combine any type of magic. Hemulurgy is Ruin's magic system, meaning Ruin even managed to break magic.
Hoids explanation to Shallan on how their are three types of people who change the world mirrors exactly what he has been doing in all the books so far.
Hoid: There are two kinds of important men, Shallan. There are those who, when the boulder of time rolls toward them, stand up in front of it and hold out their hands. All their lives, they've been told how great they are. They assume the world itself will bend to their whims as their nurse did when fetching them a fresh cup of milk.
These men end up squished.
Other men stand to the side when the boulder of time passes, but are quick to say, "See what I did! I made the boulder roll there. Don't make me do it again!"
Those men end up getting everyone else squished.
Shallan: Is there not a third type of person?
Hoid: There is, but they are oh so rare. These know they can't stop the boulder so they walk beside it, study it, and bide their time. Then they shove it - ever so slightly - to create a deviation in its path.
These are the men...well, these are the men who actually change the world. And they terrify me. For men never see as far as they think they do.
Like the third man Hoid is continually at the sidelines and watching the boulder of time move along. He is rarely directly involved and only has a small influences when he does, doing things like telling Siri a story, giving Kelsier or Wax information, giving wisdom to Dalinar, teaching Kaladin lessons or helping Shallan.
It does seem like Hoid is growing more bold in his involvement by doing things like becoming a radiant and helping the Radiants in more obvious ways or by outright throwing Wax a coin to show him how Kelsier is alive. As time has gone on he has either grown more and more confident in how to influence events or simply believes that things have progressed far enough that he has not choice but to be more fully involved. And despite all this Hoid is incredibly fearful to change things to much, as he is still significantly limited in his ability to ensure he will effect the events in a positive way.
Each shards respective Investure and Magic system mirrors their intent.
Ruin has Hemalurgy which must be used by killing and ripping out the spirit web of other living beings. In this case ruining them. Also the investure gifted is always lesser than it was originally. Ruin cannot create, and even in the process of taking something it's impossible to avoid some destruction.
Preservation has Allomancy which is investure positive, meaning it creates more investure when used. As per word of God though this converts the metal into investure, thus preserving it.
Harmony is the combination of Preservation and Ruin and thus has both magic systems. Whilst the holder of his shard Allomancy has also grown less powerful due to the bloodlines being diluted and Hemalurgy has grown less prelevant. Since Harmony has been in charge both preservations and ruins systems have grown weaker and Frechumancy the balance between them has grown more popular and prelevant.
Emdowment has Biochromatic Breath which has to be given to someone else willingly. In order to use magic, Breath must be collected. The magic system works on giving gifts to people.
Honour has Stormlight. One has to follow their oaths in order to have access to it.
Odium has yet to show his magic system. His corruption of Stormlight though works by taking away their users emotion and then giving it back later to enrage them. All of the unmade also work by manipulating people's passion and enhancing them.
Cultavation similarly has yet to reveal her magic system. However her method of giving people curses and boons work so that she never loses any investure. Instead it only grows as she cultivates it.
Odium is a foil to Ruin, the only two Shards we've actively seen who's been in the forefront as a villain. While both employ prophecy, have empowered minions, and notably managed to kill the Shard they were sharing a territory with who opposed them, both differ wildly on implementation, with Odium primarily using methods based on the Cognitive Realm and Ruin being mostly based on the Physical realm.
Ruin's own use of prophecy mainly composed of altering existing prophecies to serve his ends, sometimes moment by moment to fit the situation. Odium is noted to be very good at making his own prophecies and sees very far into the future, to the point that predicting the future is actively taboo on Roshar due to how closely it is tied to Odium. They don't even play gambling games that rely on guess the future result of, say dice rolls, because that smacks too much of prophecy, instead playing variants where they guess the numbers after dice have been rolled, since that makes it a past event.
In appearance, Odium tends towards gold and white colors, and appears as a gold and/or white clad member of the species he is appearing to. Ruin tends toward black and dark gray, and appears as someone the viewer knows. Odium tries to make the person he is talking to trust and surrender to him with a calming, Affably Evil demeanor while Ruin takes advantage of already-established trust and subverts it to his own ends.
Both are a source of Investiture and use their Investiture to power and physically alter certain minions. In this, they are actually very alike, both utilizing corrupted versions of other Shards' Investiture, with many of their minions undergoing noticeable physical changes. With Odium, however, most of these changes are natural parts of his minions biology stemming from changes to their spiritual or Cognitive aspects that trickle down to the Physical Realm. Ruin actively alters and mutilates his minions in the Physical Realm with stolen bits of Spiritual data to alter and add to their Cognitive and Spiritual parts.
Speaking of minions, they're even opposed when it comes to their minions' sanity. Odium values sane, physically whole minions because they can act rationally, follow orders, and have sufficient initiative to do what he wants done. Ruin goes for insane minions, whether originally like that or induced to be so from contact with him, because they're easier to control with spikes and won't question doing what a voice in their head tells them to do. Notably, Odium's insane minions tend to be harmless and sit around doing nothing, while capable of fighting if ordered. Ruin's minions are always violently insane, and without his direct control rampage wildly.
Both utilize forms of mind control. Odium does it by implanting Physical beings with spren/Cognitive beings aligned to him, and Ruin does it by spiking them with Hemulurgical spikes, physically opening them up to his Cognitive tampering and control. Notably, all of Ruin's forces were in the Physical realm, with him being the only Cognitive presence, while Odium, while having a sizeable force on the Physical Realm, has a significantly more massive force in the Cognitive Realm, and is also able to corrupt and control Cognitive beings.
Speaking of mind control, Odium's acts of subversion tend toward deals and pacts that he is completely up front and honest about, and will actually go through with in both spirit and letter. This allows him to take control of populations and essentially conscript whole peoples to his side by subverting their leaders, with eventual trickle down effects. Ruin, however, plays ALL sides. The leaders, the mooks, the rebels AND the heroes. By massive and subtle manipulations, everyone is unknowningly on Ruin's side even as they are impelled to try and destroy everyone else, and even when they think they're opposing him.
Before becoming Shards, Ati, Ruin's holder, was noted as being a good man who was corrupted into being the insane god that came to destroy Scardrial, and being very honest about it. Rayse, Odium's holder, has apparently ALWAYS been scheming, evil, and dangerous, yet currently presents himself as a friendly, wise and relatable guy... when he's trying to con you into dealing away EVERYTHING to him.
This also explains their difference in methods. Ati wasn't evil, he succumbed to the shard impulses, so Ruin was unable to hide them, and he used his powers to impose himself on others and strip their mind away, just like it happened to him. Rayse, however, wanted it, and got control on his shard, and this is reflected in his subtlety.
Notably, there is also their omnicidal end goals. While Odium wants to destroy the heroes and the peoples of Roshar they're defending because accomplishing he needs to get out of the pact binding him to Roshar so he can move on and resume killing other Shards, Ruin wants to commit omnicide as a means in itself, with the implication that after he's killed everyone on Scandrial, he's going to destroy the physical planet itself and move on to continue doing more of the same. For Odium it is the means to an end, for Ruin it is the end in itself.
The end of Oathbringer also reveals that the Singers associated Odium with Humanity. It's unclear if they specifically worshipped him before, but it's obvious that eventually they ended up working with Honor against Odium. Odium doesn't just want to destroy humanity for freedom, he also hates them.
Look at their Dragons, and how they control them. Ruin uses Hemalurgic spikes on Marsh, which turned him into a brainwashed Meat Puppet who physically can't move without Ruin's instruction, no matter how stubbornly he tried to resist. Odium uses the Thrill and Amaram's refusal to see himself as in the wrong to take control, which Dalinar can resist through willpower and the ability to accept responsibility. Marsh isn't morally culpable without free will, whereas Amaram only thinks he's innocent even though he still had the choice to refuse Odium.
Also, bloodlust is a double-edged sword for both of them. Ruin made Marsh more susceptible to bloodlust, but it temporarily allows him to regain control. The Thrill and the subsequent regret made Dalinar almost tip over to Odium's side, but it allowed him to atone and improve.
On another note, compare Marsh and Amaram. Marsh infiltrating the Steel Ministry is a slow burn of a Heroic Sacrifice; he's certain he'll be caught and executed, but makes his peace knowing that the information he sends out will be vital to future rebellions. Amaram thinks his life is worth more than his subordinates because he'll be needed for the Desolations. In other words, Marsh found worth in his life by giving it up for others, and Amaram thought others were a worthy sacrifice to save himself.
In Elantris, Sarene tells the other nobles that a country is more than just land or a business, its in the hearts and minds of the citizens. The idea behind a country is a combination of Devotion and Dominion; patriotism and noblesse oblige (from the government to the people) are forms of Devotion, and Dominion is the social contract that government and citizens have over each other. Thats why the Aon magic system draws the country.
Secret History reveals that someone in the Cognitive Realm can pick up and carry the objects they find there, but the Cognitive object will vanish if their Physical Realm counterpart is moved or destroyed. Some real life Offerings to the Gods take the form of food or treats are 'sacrificed' by leaving them out overnight, which are eaten the next morning because those deities would have already taken their share, magically restoring them in the process. Since deities tend to hang around the Cognitive Realm, this is a viable way of sacrificing to them.
In Oathbringer Hoid tells Shallan a story about the three kinds of people who change the world. When a boulder is rolling down towards them, the first person tries to push it down and gets squished, the second person takes credit for the boulder squishing the first person and threatens the masses into compliance with its power, and the third person carefully observes the boulder before changing its path. If the boulder is the current disaster hitting Scadrial, the three people fit Kelsier, the Lord Ruler, and Sazed.
The first person seems ineffective because he dies, but Kelsier completely turned the world around with his death. He threw himself at the boulder and rallied everyone else to follow him, and all those bodies piled up did change the boulder's path.
The Lord Ruler is the second person because of how he justifies his reign: the people need him to keep the world safe. His whole religion has an undercurrent of "I saved the world, be grateful". While he is powerful, he couldn't stop the boulder. He genuinely believes that their compliance is required to defend them from the boulder, but he's still using it to control them.
Sazed is the third person because he uses his knowledge to move the boulder more or less permanently away. Unlike Kelsier, he could see the consequences of his actions and picks the best route. However, he can't destroy the boulder, and accidentally kills a good chunk of the Southern Scadrian population when terraforming.
In Secret History, Preservation compares Kelsier killing the Lord Ruler and inadvertently freeing Ruin to saving people from a burning boat by sinking it, and then pointing out that they didn't burn to death. That's more or less what Shallan did in Words of Radiance; when assassins were killing everyone on her ship, she turns the boat into water. In both cases, they turned certain death into possible death because they didn't have a better way.