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Previous versions had rules about the total number of spell that a spellbook could include calculated using the level of the spell and number of pages in the book.
5e mentions that a spellbook has 100 pages, but I did not find anything (so far) that says how many pages (if more than 1) a spell takes in a spellbook.
Can I assume that 1 spell uses 1 page?
You're right that the 5e PHB does not mention how many pages are needed for spells, though as you point out spellbooks are given as having "100 blank vellum pages" (PHB 153).
The only hint that there might be a difference per spell level is in comparing these two texts from the "Your spellbook" sidebar on PHB 114:
For each level of the spell, the process takes 2 hours and costs 50 gp. The cost represents material components you expend as you experiment with the spell to master it, as well as the fine inks you need to record it.
(...)
You can copy a spell from your own spellbook into another book— for example, if you want to make a backup copy of your spellbook. This is just like copying a new spell into your spellbook, but faster and easier, since you understand your own notation and already know how to cast the spell. You need spend only 1 hour and 10 gp for each level of the copied spell.
We can deduce that the 10 gp cost for making backup copy is mostly the cost of ink, as you no longer need to expend material components for experimentation. As this amount is per spell level, it would suggest that higher level spells take up more ink, and therefore more space in the spellbook.
Spellbooks don’t even have to be books. In the Player’s Handbook we see that spellbook can be:
a loose collection of notes scrounged together after you lost your previous spellbook in a mishap.
Additionally, Xanathar’s Guide to Everything gives some more options for spellbooks:
Long straps of leather on which spells are written, wrapped around a staff for ease of transport
Small stones inscribed with spells and kept in a cloth bag
The idea here is that a spellbook can pretty much be anything you like; in this section of XGtE, the game encourages you to be creative:
Your wizard character’s most prized possession — your spellbook — might be an innocuous-looking volume whose covers show no hint of what’s inside. Or you might display some flair, as many wizards do, by carrying a spellbook of an unusual sort. If you don’t own such an item already, one of your goals might be to find a spellbook that sets you apart by its appearance or its means of manufacture.
So the answer to your question is “there is no limit unless you want there to be one”, because it is your spellbook. Design it how you like.
Fifth edition tries to not force you to do administration chores like tracking encumberance or counting spellbook pages used, unless you enjoy it. Therefore, there is no hard limit on the numbers of spells you can put into your book in the rules. It could be arbitrarily many if you don't care for this part of the game.
However, if you do care, we can deduce at least an upper bound for traditional spell books.
There are four statements in the rules that are relevant for the amount of space in a spellbook. First, the wizard starting equipment on p. 114 PHB and the description of the spellbook on p. 153 PHB
You start with the following equipment, in addition to the equipment granted by your background: (...) A spellbook
Spellbook. Essential for wizards, a spellbook is a leather-bound tome with 100 blank vellum pages suitable for recording spells.
As all the equipment in the class descriptions refers to the equipment from chapter 5, the spellbook that a wizard starts with has 100 pages. It is not one of the fancy alternate spellbooks you could conceivably have.
There is no mention of needing to buy additional spellbooks to have room for your spells. This means, the spellbook of 100 pages you start out with must provide sufficient space to write all the spells you learn up to level 20.
Secondly, we also know that the ink a spell consumes is proportional to its level (page 114 PHB, also mentioned in the accepted answer):
You can copy a spell from your own spellbook (...) You need spend only 1 hour and 10 gp for each level of the copied spell.
Whatever the unit of space is, as the ink used to scribe it is proportinal to the spell level, the space the ink uses should be proportinal to the spell level, too.
Thirdly, the statement about adding spells to the book (p.114 PHB) and the wizard table tell us how many spell levels need to fit into the book:
At 1st level, you have a spellbook containing six 1st-level wizard spells of your choice. (...) Each time you gain a wizard level, you can add two wizard spells of your choice to your spellbook. Each of these spells must be of a level for which you have spell slots, as shown on the Wizard table. On your adventures, you might find other spells that you can add to your spellbook (see the “Your Spellbook” sidebar).
The additional spells from adventures that can be copied into the book are optional, so we will ignore them. The spells gained from leveling are not. The upper bound of spell levels you can add to your book for a wizard are initial 6 first level spells from level one, plus two times the levels of the highest level spell you can learn in the wizard table for each additional character level, all the way up to level 20. This totals up to 184 spell levels. All of which will need to fit into your book.
With all of this, we can put an upper bound on the pages a spell level can take up: it must be possible to fit 184 spell levels into a book of 100 pages. That means, at most a spell level can take up a little more than half a page (54.35% of a page).
If you went with the one page per level rule instead, you could expect that at the latest by spell level 7 the wizard would need to buy an extra book. If they find a good number of additional spells, much earlier (we used that rule and I had to buy one around character level 8).
The rules do not provide a lower bound on how many pages a spell needs. You could go down to small fractions of a page -- or you could forgo traditional books and pages entirely (see Thomas's answer), if you are so inclined.
Purely to give some reference points: there are 829 levels worth of wizard spells in the PHB so to allow a wizard to collect all the spells from the PHB without needing a second book, each level could only take up less than an eigth of a page.
Considering that a wizard might find some but not all additional spells adventuring, if you even care about tracking pages, a good number could be half a page per spell level. This would leave room to add another 16 levels of spells to the book, without having to buy an additional book.
No core mechanics in 5th edition in regards to page usage. As mentioned above, the section of the PHB with your spellbook's information/appearance allows you to use anything that you could record information on, at the cost of 2hrs and 50gp (per level of spell) for new spells, and half as much for ones you already have prepared/recorded.
In the AL module 'The Visitor,' the antagonist's spellbook was a collection of brass plates/sheets, attached together by a large brass ring. Your entire spellbook could be a gilt tome, a loose collection of notes, a leather volume, etc. The point is that you can make it as simple, flourished/flavorful as you like, as per the core mechanics.
If the DM wishes to put any additional costs to learning spell/skills, that's between them and their players to figure out. As far as Adventure's League goes, it is my understanding that you must use the information that is openly available as per the PHB/ errata for that edition. If it's not mentioned, then it has been excluded from that edition, (similar to what happened with paladins and monks between the different editions)
I could find no explicit rulings on this within the released material.
pp 114 PHB for spellbook description particularly the section called The Book's Appearance in the bottom left corner.
Now I don't have any wizards in my current party but I'd DM it as being a magical collection of the characters known spells and let them decide from there.
Their decision hardly affects the game, maybe they prefer to keep all their spells written in chicken scratch on a notecard like a cheat sheet from school that they squint and read off. Or maybe its a massive tome with a whole entry for every spell including its history, origins and detailed uses throughout history.
My personal ruling on the matter has always been and remains the same: each spell takes up a number of pages equal to its spell level.
There are three reasons I do this:
The first reason is simply that it's important to have a simple solution to a simple question — how many spells will fit in my spell book?
The second reason is to create an economy in my game setting that includes the wizard. Wizards in my game fill up their spell books pretty quickly, so it keeps the wizards participating in the market with buying spell books. At 100 gp a pop, that's nothing for a high level wizard to afford.
The third reason is that, now that the wizard is in the market for a new spellbook it gives me the opportunity as the DM to homebrew spellbooks with metamagic abilities worked into the pages, and the wizard attuning to a different book when they need to fill a different role. Here is an example:
Metamagic: Maximize, Tome (Very Rare: 50,000 gp), Requires attunement
100 pages (blank). Spells cast from the spellbook are empowered with metamagic ability. Maximize: Spells cast with this metamagic ability deal the maximum damage for the spell level it was cast.
