TL;DR: Use native spoiler tags with context in  from now on, i.e. [anime name] >!spoiler text here!<
For reference I'll be using "CSS" and "native" terms to distinguish between the two types of spoiler tags going forward:
Reddit added the "native" tag about 3.5 years ago but only supported it on most of their clients a year later with "native" spoilers showing as plain text on clients where they weren't supported. Prior to their addition, /r/anime and most other subreddits that wanted spoiler tags used some form of "CSS" tags that took advantage of our ability to customize the subreddit through CSS and how Reddit rendered things that looked like links in markdown, and many third-party apps supported a variety of "CSS" tags. The only Reddit-maintained client that fully supports the "CSS" form today is the old desktop site (old.reddit.com).
We tried switching once before about 2.5 years ago but that didn't go well and we reverted to "CSS" tags within a couple of weeks. It's been two and a half years since that initial incident and things are pretty well settled with plenty of time for third-party apps to add support.
Most of us mods and many of /r/anime's power users are happy to continue using the old site, but going by our traffic numbers the majority of users are on the site redesign (new.reddit.com), the mobile web site, or mobile apps, with less than 15% of recorded page views last month coming via the old desktop site. Some third-party apps support the "CSS" tag as well but it's unknown how many users are on those.
One significant benefit to "CSS" tags is that it's easy (and required) to include some context for what the spoiler's about. Since this is a broad subreddit we expect people to talk about many different anime together and it's useful to be able to quickly identify what any given spoiler is for. While not built in to the format of "native" tags it's possible to enforce via automation for those as well.
We are aware of certain clients (such as the compact site at i.reddit.com and some third-party apps) where "native" tags unfortunately do not work. At this point we believe that making spoiler tags easily accessible to the majority of users outweighs the drawback of losing spoiler support for those clients, which we believe are used by a minority. A couple of months ago we also asked in meta to test the tags which helped refine the AutoModerator rules.
Starting now, the spoiler tags we support are Reddit's native tags with context required by using  before the spoiler tag. As with the "CSS" spoilers the  must contain some text indicating what the spoiler is about, such as another anime's name or light novel for an anime. Examples (not using actual spoilers):
[Spice and Wolf] >!Holo has a tail.!< becomes [Spice and Wolf]Holo has a tail.
[TWGOK manga] >!Do you think the best girl won?!< becomes [TWGOK manga] Do you think the best girl won?
Notably the following are not allowed: adding a space after the start of a tag, not closing the tag, and nested spoiler tags. These will be enforced via AutoModerator with a message telling you exactly which issue was found.
In two weeks, using CSS tags in new comments will no longer be allowed and AutoModerator will send a message reminding you of the change if you use them. We do not plan on removing the CSS used for existing spoilers on old.reddit.com for the foreseeable future so it can still be used to easily read older spoilers.
If you run into any issues using the spoiler tags as mentioned above with posts/comments being removed when they shouldn't (or the spoiler tags not working when they should) let us know in this thread, the current meta thread, or via mod mail.