With Modern Horizons having roughly eight billion different premium treatments, along with multiple booster types, it can be difficult to figure out what you can actually expect out of any given box.
We're operating on two assumptions here, based off of previous Booster Fun treatments:
A showcase card will replace the regular version one in three times that card shows up.
A card with multiple showcase or booster fun treatments will still show up at the same rate and with the same showcase frequency as one with fewer treatments. This means that a card with two showcase treatments will have each specific treatment replace one in six times that card shows up.
Draft Boosters, Regular Rares/Mythics: There are 60 rares and 20 mythics in the set. Every rare should show up twice as frequently as every mythic, meaning a total print run is 140 rare cards, with specific rares showing up 1 in 70 packs on average and specific mythics showing up in 1 in 140 packs on average. Given 36 packs to the box, this means that you can expect to open roughly half of the normal rares and a quarter of the mythics (30 and 5, with the 36th likely being an additional rare) in your box. This also means you are likely to open 10 rare showcase (retro/sketch) and 2-3 mythic showcase (retro/sketch/borderless) in your box. The expected number of Fetches per box is 36*(5/70), or about 2 and a half. That means if you open an entire case of 6 boxes (I'm assuming 6 boxes to the case here but some supplementals have had 4), you should open 3 of each fetch on average, with one of each fetch on average being Retro Frame per case.
Draft Boosters, Reprint Rares/Mythics: The way the new-to-modern reprints work out, a complete print run across 140 packs will have 1 of each mythic, 2 of each rare, and 4 of each uncommon. That means in a 36-pack box, you can expect to get approximately half of the 18 rares (9) and one quarter of the 4 mythics (1), plus one of every uncommon.
Set Boosters: Set Booster boxes normally work out to get approximately the same number of main-set Rares/Mythics as Draft Booster boxes, despite having only 30 packs vs the normal 36, because of the wildcard slots. This includes fetchlands, which should also show up at a rate of ~2.5/box and 15/case of six boxes with 1 of each per case in retro. However, given that Set Boosters only have one Reprint slot per pack, the chances of opening a specific reprint Rare/Mythic in a set booster box are about 17% lower.
Collector Boosters: These are the most complicated booster, as there are four different rare slots and each has as different set of available cards to pull from, so I'll cover each slot individually.
Etched Foil Rare/Mythic: There are 61 possible rare options from Retro MH2 rares (31), Retro MH1 reprints (12), and New to Modern Rares (18). There are 16 possible Mythic options from Retro MH2 Mythics (7), Retro MH1 Reprints (4), and New to Modern Mythics (5). This makes the full "print run" 138 cards, which is just a hair smaller than the 140 cards in the normal rare print run. That means a specific rare should show up in about 1:69 Collector Boosters (5.75 boxes) and a specific mythic will show up 1:138 Collector Boosters (11.5 boxes). As there are 5 fetchlands, there should be an Etched Foil Retro Frame Fetchland every 13.8 Collector Boosters (i.e. 86% of Collector Boxes should have one).
Foil Special Frame: This includes all Retro Frames, Sketch Cards, Borderless Alt-Arts, Extended Arts. Every main set rare and mythic has at least one of these treatments. On top of that, there are four new to modern reprint rares with borderless treatments, and four mythics with this treatment as well. Additionally, 2 rare and 1 mythic new to modern reprints get old borders -Titania, Shardless Agent, and Chainer - but of those, only Chainer does not also have a Borderless treatment. Finally, there's 12 rares and 5 mythics from MH1 with Retro Frames. All told, this means 60+4+1+12=77 rares and 20+4+5 = 29 mythics. With two of each rare to each mythic, that makes a total print run of 183. That means a specific rare will show up in approximately 1:91.5 packs (7.625 boxes) with any one of its foiltreatments. Rares with two treatments will have a specific treatment show up 1:183 packs (15.25 boxes). A specific Mythic rare will show up with any treatment 1:183 packs (15.25 boxes). Mythic rares with two treatments will have a specific foil treatment show up in 1:366 packs (30.5 boxes). The chances of pulling any foil special-treatment fetchland is approximately 1:18.3 packs (1.525 boxes), or approximately one in two thirds of all boxes, with a third of boxes having a foil retro fetchland and a third having a foil extended art fetchland.
Extended Art: There are only 39 possible options for this slot, all at rare. Five of those are fetchlands. This means 5:39 packs, or a little better than 1:8 packs will have an extended art fetchland. This means you can expect 1-2 extended art fetchlands per box, with about half having 1 and half having 2. The chances of opening a specific rare are 1:39, or about 1 in 3.25 boxes. All 39 of these cards are main set rares, with none being new-to-modern reprints, MH1 retro reprints, or mythic rares.
Sketch or Borderless: This slot is a little odd as it does not have any chance of being a Fetchland or a MH1 retro reprint. There are 4 new to modern rare reprints and 4 mythic rare reprints. 19 of the 20 mainset mythics have a borderless treatment. 8 of the mainset mythics have a Sketch treatment - the only overlap between these two sets are the 3 planeswalkers, which means there's all 20 possible mythics available from this slot. There are also 21 main set rares with the Sketch treatment. This makes all the available cards 25 rares and 20 mythics. However, this is likely not in a straight 2 to 1 rare to mythic ratio, as that would mean this slot is mythic a little less than a third of the time. It's more likely that there's 4 or 5 of each rare to 1 of each mythic, for a print run of 120 or 145. Because of the uncertainty here, it's impossible to give a specific ratio of how often each card will show up in this slot.
All Together: If you are specifically looking for Fetchlands, Approximately 1:4 Collector Boosters should contain one of any fetchland in some treatment, or about 3 per box.
Hopefully this post helps to inform your choices as to what boxes to buy if you're looking to do so.