Death's Shadow is currently considered as one of the top-tier decks in Magic: The Gathering's Modern right now, so be ready to face it at your upcoming tourney. It comes in many variants such as Grixis, Jund, 4-Color and sometimes pushing it to 5-Color. Despite the variations, all these decks boil down to a core game plan: Disrupt the opponent's hand, gain information of opponent's game plan, stick a threat, use cheap removal to remove blockers. Beatdown with threat.
In no particular order, here's a list of sideboard cards to help beat Death's Shadow in Modern.
Rest In Peace
Pros: This card is an all-star versus Death's Shadow. It slows down casting of Delve creatures, turns off Delirium, Tarmogoyf becomes a plant and turns Snapcaster Mage into an Ambush Viper. Cons: Drawing this in the mid-to-late game maybe a little too late. In capable hands, the opponent may have already resolved a threat and is about to close the game with the beatdown. Death's Shadow can still provide the beatdown with its huge power/toughness and Delve creatures can still be hard-casted.
Leyline of the Void
Pros: This card is bonkers when it's on the opening hand. Similar pros as Rest In Peace. Getting rid of it from the battlefield will be difficult for the opponent. Cons: Drawing this card is simply awful. Having to rely on luck to have it on the opening hand is an iffy strategy.
Relic of Progenitus
Pros: Proactive use of this artifact will definitely slow down the opponent's game plan.
Cons: After using the "exile all graveyards" ability, the opponent can simply rebuild their graveyard again.
Nihil Spellbomb
Pros: One-sided cleanup of opponent's graveyard. Cons: One-time use only and opponent can rebuild their graveyard again.
Mirran Crusader
Pros: An absolute all-star in both offense and defense. Majority of the creatures from the Death's Shadow decks are black or green, namely: Death's Shadow, Tasigur, Gurmag Angler, Street Wraith and Tarmogoyf to name a few. It also has protection from Fatal Push and Terminate.
Cons: Easily dies to Lightning Bolt and Tarfire.
Chameleon Colossus
Pros: Similar to Mirran Crusader, it dodges black spot-removal and black creatures. It can quickly close out games given the opponent's low life total.
Cons: Given the right boardstate, it can be outmatched by a bigger Tarmogoyf and can still be answered with an edict effect.
Leyline of Sanctity
Pros: This shuts down the hand disruption plan and let's you execute your game plan. Cons: Similar to Leyline of the Void, horrible when you draw this card and excellent when it luckily lands in your opening hand.
Chalice of the Void
Pros: Counters majority of the cards in the Grixis shell including Death's Shadow itself. Usually putting the charge counter to 1 is enough to slow down majority of Death's Shadow decks. Cons: Prone to Artifact hate and Abrupt Decay can still destroy this card.
Thalia, Guardian of Thraben
Pros: A chunk of Death's Shadow decks are noncreature spells. With Thalia in play, it'll be taxing for the opponent to resolve majority of their spells.
Cons: Prone to cheap removal spells and doesn't block well against the Death's Shadow creatures.
Thalia, Heretic Cathar
Pros: This creates a tempo advantage since Death's Shadow decks pack at least 12 fetchlands. Somewhat similar to Thalia, Guardian of Thraben, it'll be taxing for opponents to cast their spells.
Cons: Prone to cheap removal spells and doesn't block well against the Death's Shadow creatures.
Leonin Arbiter
Pros: Death's Shadow decks have a lot of fetchlands and the green-based variants pack Traverse the Ulvenwald to tutor their key threats. Leonin Arbiter denies the searching abilities and heavily taxes the opponent. Cons: Prone to cheap removal spells and doesn't block well against the Death's Shadow creatures.
Aven Mindcensor
Pros: Cracking that fetchland and casting Traverse the Ulvenwald will be tricky for the opponent. Cons: Thanks to luck, opponent may still find the card they need in the top four of their library.
Auriok Champion
Pros: Auriok Champion can hold down the fort thanks to her protection. A great blocker and dodges all forms of spot removal from the Grixis and Jund variants. The lifegain is an incremental advantage especially when it comes to racing.
Cons: Her measly 1 power and 1 toughness doesn't make her much of a threat on the battlefield.
Etched Champion
Pros: 1 or 2 of these usually come off from the bench of Affinity decks. Excellent for blocking and a beast when attacking with an attached Cranial Plating.
Cons: Putting this on the battlefield without Metalcraft can just wreck this card. Most Death's Shadow decks are now packing Kozilek's Return in the sideboard just to battle this card.
Wurmcoil Engine
Pros: This creature is such a headache once on the battlefield. Even when it dies, it still leaves two 3/3 tokens that need to be dealt with.
Cons: Not much of a downside except if the opponent is packing a lot of removal that can easily answer this creature and the tokens.
Kitchen Finks
Pros: A recurring body that cannot be easily removed by the opponent. Usually saves two turns and helps you at least gain 4 life.
Cons: Primarily only used for blocking purposes, rarely attacks when Death's Shadow creatures are on blocking duties.
Hangarback Walker
Pros: Death's Shadow decks have a hard time closing out games that have decks that go wide. This creature is a resident of value town. Even after dying, it generates a plethora of blockers that can help stall the board.
Cons: Casting for only 1 counter is not sufficient enough, best to cast mid-to-late game.
Lingering Souls
Pros: Lingering Souls does exactly the heavy duty of providing blockers and evasive offense. Can easily close games in the mid-to-late game. Cons: It's 1/1 body can't exactly deal lethal damage to the big fatties of Death's Shadow decks.
Empty the Warrens
Pros: Much like Lingering Souls, going wide will help. Countering this card is not effective thanks to its Storm ability.
Cons: It's 1/1 body can't exactly deal lethal damage to the big fatties of Death's Shadow decks.
Blood Moon
Pros: Death's Shadow decks pack a lot of fetchlands and shocklands. A resolved Blood Moon can just wreck your opponent's day. Mana denial can simply help you execute your game plan while they watch helplessly. Cons: Casting in the late game may not be relevant and some opponents will fetch basic lands playing around this Enchantment.
Blessed Alliance
Pros: Letting the opponent gain 4 life can shrink the Death's Shadow and mess up combat math. Death's Shadow decks pack only a few creatures, the majority of the time it will only attack with one fatty making Blessed Alliance's edict effect worth the trade.
Cons: This card is highly situational and skilled opponents will usually play around it.
Ensnaring Bridge
Pros: Thanks to the big power of the Death's Shadow creatures, this card will put the offense at bay. It'll help you execute your game plan while the opponent watches.
Cons: You have to properly manage the cards in your hand to ensure the creatures don't attack you. Ensnaring bridge is still prone to artifact hate such as Kolaghan's Command and Abrupt Decay.
Path to Exile
Pros: Of all the spot removals available in Modern, Path to Exile is the cleanest answer to all of Death's Shadow creatures. Exiling is more effective since the creature won't hit the graveyard to help enable Delve and Delirium strategies.
Cons: Not much of a downside except for giving the opponent a basic land.
Detention Sphere
Pros: If you're lucky enough, you can catch 2 or more permanents with this card. Hard to remove once resolved because of less Enchantment hate. Cons: Green-based variants can still answer via Abrupt Decay or tutor for Reclamation Sage.
Dusk / Dawn
Pros: Death's Shadow offense are usually at an average of 4 power, this one-sided wrath effect of Dusk can just be backbreaking for the opponent. While Dawn helps you rebuild your board by going wide with creatures.
Cons: Similar to Blessed Alliance, this is very situational and a skilled opponent can play around this card and not commit too much of their fatties on the board.