Since the original MARVEL SUPER-HERO SECRET WARS back in the mid-1980s, big line-wide event stories have been a staple at Marvel. They've been more and less frequent, but they've never really gone away - and it seems unlikely they will, at this point.
My understanding is that Marvel has usually left it up to each book's team as to whether they want to participate, at least on the truly line-wide stories. There are exceptions, of course: the 2015 SECRET WARS basically paused every Marvel book.
However, participating in a cross-over event like these can be a way to at least get a temporary sales bump. Very few new readers may stay past the crossover (sales rarely go up permanently), but there's definitely an incentive to get involved.
Now, a title's participation in an event can range from the critical (the event results in a long-term change to the book's status quo - for example, the 1990s "Fatal Attraction" X-Men event, which resulted in Wolverine's adamantium being removed from his skeleton (for a couple of years' worth of stories, at least) to a "red skies" crossover (a term from DC, actually - during the CRISIS ON INFINITE EARTHS crossover series in the mid 1980s, some things that were labeled as "crossovers" simply involved a character noting weird weather related to the story, with no other connection - and no interruption of that book's on-going storylines).
In really major events, the main story is most often told in a separate series (AVENGERS VS X-MEN, AGE OF ULTRON, SECRET WARS, CIVIL WAR (both), and most recently SECRET EMPIRE. If an on-going title ties into the event, it's usually not a significant part of the overall story, and is meant to stand alone - but, it still won't make a lot of sense, if you don't know what's happening in the main story.
The worst, from your perspective, are the "family"-level crossovers, like X-MEN: BATTLE OF THE ATOM, AVENGERS: OPERATION GALACTIC STORM, or the ALL-NEW X-MEN/GUARDIANS OF THE GALAXY crossover stories, "The Trial of Jean Grey" and "The Black Vortex".
(Note: I will reference significant developments from the above comics below - scroll past the bullet list below to avoid.)
In these cases, the main storyline did indeed wend its way through each book involved. Just reading consecutive issues of one book would mean that you'd skipped 4 chapters of the story, and things wouldn't make a lot of sense.
For these, if you don't have enough of the story to follow them, then my best advice is, "Don't". Just skip over those issues and pick up after the crossover's done.
This can be problematic, however. Looking at three the four examples I listed above shows at least one notable on-going changes to some books' status quos, or sub-plots that arose and continued on for some time:
- "Operation: Galactic Storm": At the end, the Avengers discuss whether the architect of the story should be killed. Most of the team doesn't think so, but several characters (including Iron Man and the Black Knight) do - and they act on their decision. This causes (or exacerbates) a rift between Cap and Tony Stark, leads to some minor shuffling of the team, and leads Cap to leave the team for a couple of years. It also causes issues for Dane Whitman (the Black Knight). I believe that Wonder Man had some problems related to an explosion he was subject to during the war, which was covered in his on-going series).
- "Battle of the Atom": this impacted three of the four books involved (ALL-NEW X-MEN, UNCANNY X-MEN, and WOLVERINE AND THE X-MEN, with Kitty and the time-displaced original X-Men leaving Wolverine's school and joining Scott's. Also, the future X-Men seen in the story that were left in our time reappeared in ALL-NEW X-MEN later.
- "The Trial of Jean Grey": Kitty wound up dating Peter Quill (Star-Lord), and young Cyclops left ALL-NEW X-MEN to spend a year traveling with his dad (in his own book).
That said, these changes generally aren't just covered in the title they affect during the crossover. You will feel like you missed something - but trying to read parts 2, 9, and 16 of a story is not likely to help you.
One caveat - if the crossover ends in the title you are reading, it's probably worth reading that. Yes, you won't quite know what's lead up to this ending, but you will know what's coming out of it at least.
Now, you actually asked about avoiding this altogether. That's hard to do, except in retrospect. Any on-going book that meets these criteria today may not continue to do so. That said, here's a few that would work now:
- MOON GIRL AND DEVIL DINOSAUR
- UNBEATABLE SQUIRREL GIRL
- UNBELIEVEABLE GWENPOOL
and a few that are currently safe, but have a history of having been involved in these sorts of things in the past:
- MS MARVEL (in that her origin came out of the line-wide "INFINITY"/"INHUMANITY" crossover storylines)
There are definitely others - these are off the top of my head.
Now, looking back into the past, there are definitely some relatively safe titles. Again, these are off the top of my head, not an exhaustive list:
- ANTMAN/ASTONISHING ANTMAN (2015-2016)
- IRREDEEMABLE ANT_MAN
- SPIDER-MAN 2099
- YOUNG AVENGERS (any run)
- RUNAWAYS (any run)
- IRON FIST (1970s)
- Grant Morrison's NEW X-MEN run (early-mid 2000s)
Some titles are less affected by this sort of thing than others. The POWER MAN/IRON FIST run from the 1980s had at least one story that crossed over from another book (and since that book was ROM, which Marvel doesn't have the rights to, you probably won't find this anywhere but back issue bins, if there), and at least one SECRET WARS II tie-in (but that doesn't really require reading anything else) - but I think that's it.
Basically, (as noted in Jack's comment to your question), characters that aren't well-known, and that aren't intimately tied into other books by dint of team membership, character history, etc. are your best bets.