The sideboard to this deck can be completely different iteration to iteration and is solely dependent to what is in your meta. I am going to go over the most popular choices and possible some fringe cards I have seen in the past, noting my personal success with each of them.
Engineered Explosives
This card is a powerhouse. Putting it on 3 against a blood moon, Liliana of the veil and even aven mindcensor is amazing for feeling safe, and then blowing out token decks or even lingering souls is well worth it. Card is very flexible and very useful. Also, a cool trick with it is if a Thalia, Guardian of Thraben is in play, you can spend UWR to play it for 2 and then it comes in with 3 counters. The same works on 2 and 1 as well. If you want to avoid this and play an EE for 2 with 3 mana, just announce that you are paying with UUX or something like that.
Ancestral Vision
I used to play this card in the main of UWR Nahiri, and overall the card can pull you ahead of Jund/Junk, but other than that all tihs does is rot in exile or in your hand when you play your bad matchups. Similar to Spell Snare, I think this only makes your good-50/50 matchups more in your favor. I do not like playing this in the control shell because I think it is too slow for 70% of the field. Also, the sideboard cards should really help out with the tougher matchups, so I do not think this card deserves the slots.
Dispel
This is the counterspell to win counterwars, stop instant speed combos, and fight some removal/general powerful spells. It is primarily a card to fight against other blue decks as cryptic command can be a little bulky, but stopping a chord of calling, collected company, ad nauseum or even something like a path or fatal push on your colonnade is very good for a blue mana.
Card:Ceremonious Rejection
I have recently been using this for Eldrazi Tron because it counters 100% of the spells in their deck. I have given up trying to mess with their lands as it does not seem like a worthwhile strategy in this deck, so having this as a counter to a Chalice of the Void, Thought-Knot Seer, Karn Liberated, Ugin, the Spirit Dragon and Ulamog, the Ceaseless Hunger is beyond useful and has an immense of value.
Condemn
Path to exile #5. When the current meta is full of abzan, jund, death's shadow and eldrazi tron which lightning bolt does not do very well against this card is a fantastic option for the board and possibly main. Also, if you condemn a death's shadow player's Tarmogoyf or Death's Shadow, then they gain life setting their plan back and potentially killing their other Death's Shadows.
Wear / Tear
There are few very consistent sideboard cards I see and this is definitely one of them. The ability to be a 2 for 1 for 3 mana is huge when you hit something like Eidolon of the Great Revel, Blood Moon or Cranial Plating out of affinity (happened in the semifinals of a pptq for me) and even then the flexibility you get out of this card is well worth the 2 slots. Just remember when you are playing a blood moon deck to search out your plains relatively quickly.
Celestial Purge
This card has been in every single control deck that plays white for me because of 4 cards: Blood Moon, Liliana of the Veil, Koth of the Hammer and Keranos, God of Storms. All of these threats if they resolve are a nightmare to deal with. This works very well as a one for one with cards that are supposed to get opposing strategies a ton of value. It also just serves a purpose as a definite removal spell against any black or red deck, it is another kill spell against Burn for example. I would highly recommend always playing at least 1.
Stony Silence
While I definitely bring this card in against affinity, the 2 main decks this card is for are Tron and Ad Nauseum. By shutting down the draw and filter effects in tron and the mana bloom acceleration in ad naus, the matchups become way more winnable. You are essentially turning these decks into fair decks which they are not good at playing. I have had huge success withi this card, but if the meta is not heavy with these 2 then you can cut them because the affinity matchup is already so good preboard.
Rest in Peace
I have seen decks Run Rest in Peace which is why I am including it, but one of the best ways that we get value is through Snapcaster Mage, Torrential Gearhulk and Think Twice, so this hurts us a decent amount. The only reason I can see runnning this in your sideboard is if you are in a meta that is very heavy dredge. Outside of that I think Relic of Progenitus, Anger of the Gods and even Tormod's Crypt are better answers for exiling.
Runed Halo
While I have not had personal experience with this card, I have heard a lot of good things about it as it can be used against some of the hardest cards to deal with like Death's Shadow, Grapeshot, Lightning Storm and Valakut, the Molten Pinnacle, so it has the potential to be a fantastic card in the board.
Vendilion Clique
Another card solely for the combo matchup, UWR is not good at interacting with opponents hands. The deck prioritizes dealing with on board threats, but for decks like storm, ad naus or scapeshift, if they are just sculpting a perfect hand to win a counter war you are going to lose. I like playing one, maybe 2 would be correct if you are in a combo heavy meta of if you play a more ETB heavy version.
Geist of Saint Traft
Geist is primarily for winning matchups where you need a clock. The worst matchups for this deck are other, bigger control decks like UW As Foretold, any variant of Tron, card:Ad Nauseum, Storm, Esper Control or even the more pure UW control deck. Supreme Verdict and/or creature removal is pretty bad against our deck so a lot of it will be coming out, and no one in their right mind will bopard in wraths, so Geist can come in and clock someone 3 times and the game is over. We turn into a tempo Delver of Secrets
style deck with a very fast clock.
Izzet Staticaster
I think this card is amazing because it single handedly wins the affinity and infect matchups while having random game against coco decks and lingering souls decks. Your deck can handle bigger spells like Primeval Titan and Collected Company very well, but sometimes the little guys sneak through. Staticaster helps you save your Path to Exiles for better cards.
Gideon of the Trials
While I have not seen this in any side board yet, I think it could be good for the non interactive combo decks that run around the field, the first one coming to mind being card:Ad Nauseum. This can be an "I win" button against decks that can't deal with it and a solid stalling machine against decks that try to load up on one creature.
Timely Reinforcements
I will never play without one of thees in my sideboard. This is the most essential card to have against burn and other fast aggro decks. The life gain is incredible and the soldier tokens can also double or triple block a creature to make it a 3 for 1 at some points. I am currently on 1 but I can see an argument for 2 depending on the number of Lightning Helixs you have and how aggressive your meta is.
Anger of the Gods
I currently have 2 in my board to deal with both the coco combo decks and dredge. I have found this card to be very effective over something like Relic of Progenitus because it keeps my graveyard in tact while removing the main threats both of those decks have to offer. If you have the slots to play both then that might be best because they have different strengths in a number of different matchups. This also helps with the elves/affinity/zoo matchups by just being a good 3 mana sweeper.
Crucible of Worlds
This card might be low impact in some matchups, but I think giving yourself some inevitability and card advantage against decks like Emeria, UW control and Jund/Junk is worth the slot. This card also comes in against Tron as a way to keep Looping Ghost Quarter, which has proven to be very good. This is definitely one of the flex slots though, so adhere to your meta.
Jace, Architect of Thought
This is the other card that fights Lingering Souls, but it is also good against Jund/Junk where you want your topdecks to just be bombs. He draws cards in a pinch and he stalls the board by making everything smaller when things attack. He is good for token strategies, low to the ground strategies and grindy games.
Crumble to Dust
This is the best tol we have in a tron heavy meta. This card is something that they are terrified of because IT WORKS!! I have won games against the deck solely by resolving this and then just playing a better control game. You can also side this in against Jund, Junk, Grixis, UWR/UW, Scapeshift and any other deck with noteable lands/manlands and have it be great. Overall just a solid card.
Wrath of God
This card is an allstar if you ever find yourself playing elves or even across from this deck's worst enemy, Thrun, the Last Troll. These regenerating threats are miserable to deal with, but something like this is an option that is pretty universal. You could play it in the main or side, but it feels more like a sideboard card to me because there are not a lot of G1 regenerating threats you will have to deal with and if it gets countered against a delver deck you are in a terrible spot.
Hallowed Burial
This card kind of gets outclassed by Anger of the Gods because it comes down a lot earlier and is generally more efficient. It is better in bigger metas, but I think overall it is just a little too slow for the format, but lists have played them and it works against cards like Thrun, the Last Troll, Wurmcoil Engine and persist creatures like Kitchen Finks.
Elspeth, Sun's Champion
This is a very common win condition to have either in the main or in the side, but in the side it basically comes in against any topdecking game. Jund/grixis/abzan hate seeing this card, and it also givesyou a clock when you need it. Oddly enough I've brought this card in against storm and ad nauseum before just as another way to actually win the game. I don't think it is bad in the main but I am definitely going for a more instant speed version of the deck so She does not fit for my G1.