We get a small amount of SEO pagerank garbage and phishing attacks (most of which don't actually work because we don't support HTTP POST). We have user reporting and detection code that catches basically all of it, the scammers get sick of getting banned within a few hours (on average) and leave to find a host that's... not as good at this.
Our policy is to remove phishing and spam and report it to stop spam services. The major reason for this is that I want to keep our nose clean for pagerank purposes. I don't want Google to decide we're just a garbage dump and dock all our sites in search results. I surmise (rather, I hope) that our pages have a much higher rank than many other free hosting providers (and even some major social networks) because we work so hard to be proactive on this problem.
I wish Google provided better resources for this than they currently do. I'd love an API to report SEO pagerank spam, for example. Right now I have to do it manually with a form.
Abuse was not a reason, it was the reason we ended up rolling our own anycast CDN with our own ASN (https://whois.arin.net/rest/asn/AS395409). Abuse reporting goes to whomever owns the IP address, so our hosting providers would get these complaints. There was a real concern we would lose our hosting providers. The important thing to understand is that most hosting providers think you're just running a Wordpress blog, so if there's an abuse complaint they just think your blog got hacked and treat you accordingly. If you want to be an ISP, learn BGP, the cloud will not save you.
We don't get many DMCA takedowns, it's usually something really dumb and arbitrary. One of the reasons for this is probably that we blacklist certain types of file formats (such as videos and mp3s). The idea with Neocities is to use it to make web sites, so for free sites, we only allow file types useful for making web sites. Supporter accounts are not subject to these restrictions.
We used to get more classic spam (porn, "dick pillz", etc.) but not as much these days because it isn't as effective. That spam tended to hover around whatever social ill Americans were experiencing. For example, before the Affordable Healthcare Act kicked in, we got a lot of online pharmacy drugstore spam preying on people that couldn't afford their medicine. It might come back if they start dismantling the ACA.
As for comment spam (for the main site), we have strict requirements to allow commenting. You have to wait a week, make a site, update it at least a few times, and do some other specific things before you can comment.
That's not only helped with spam, it's also helped to deal with jerks. It's hard to stay mad enough at something to wait a week while engaging in psychological sublimation, just to leave a turd on someone's site profile at the end of it.