On Stack Exchange we have found that in general (though not exclusively) users with throwaway e-mail addresses are not the sort of users you want on a serious, professional site. While there may be legitimate reasons for some people to use them, the vast majority of such users aren't interested in interacting with the community.
There is also a very serious issue with such accounts - it's all too easy for someone else to gain control of a throw away e-mail address, so it makes it easy for those with nefarious intent to take over your account. This is so serious that Stack Exchange have stopped sending out recovery e-mails to such addresses:
Mailinator (and other services like it) allow you to create a fully disposable email address. What this also means is that anyone could change your password on your behalf.
This isn't a theoretical possibility. A few months ago someone did just that for a bunch of accounts on Stack Overflow. (Fun fact: people who are inclined to sign up for sites using mailinator are also fond of using their username as their email address.)
Why isn't the password recovery system automatic for certain email domains?
So, a the very least I'd implement the same restriction. Allow people to sign up, but if they lose their password that's it. This is effectively option b.
Depending on the service you are offering, I'd probably seriously think about option a - not allowing signups from such addresses. Obviously you'll have to keep your blacklist of domains updated as new temporary email services started up, but hopefully that shouldn't be too onerous.
As for people wanting anonymity - as long as you don't display links to social media unless people opt in and allow "fake" names then there shouldn't be an issue here.