OpenID is a great concept: to help you create a single identity that works across a range of Internet sites. However, it has been falling out of favour in the past few years because both Facebook and Google have provided their own replacements for it. The reasons for these are supposedly for user convenience. However, the fact is that this convenience comes at a cost. It enables Facebook or Google to collect more information about you for their commercial purposes. It also has the downside that it gives information about you to the third party website, and in some cases a rogue site can use it to post to your social media account without your permission.
Facebook's site originally supported OpenID, but some time ago they replaced it with Facebook Connect, the API that allows you to sign in with Facebook credentials on third party sites. This is good for Facebook that they can drive people to get a Facebook account that they can also use to log into sites. However people have found they are giving too much information and in some cases security privileges to the sites that receive the oAuth token from Facebook.
I wrote this post when I learned that Blogger had dropped support for OpenID. I have never used OpenID for anything at all, but now there won't be much point in using it if all these sites are going to drop it.
The primary concern I have with using Facebook or Google credentials to log into a site is that it represents the same concerns about using a single username and password to log into a lot of sites. Most of the time this will not be an issue, but if there is a problem with your Facebook or Google login, you may find that your access to third party sites is restricted or blocked.
My own experience is of inappropriate misuse of information released from a social media profile. I remember signing up to the Stuff website with my Facebook credentials. Nek minnit the Stuff website was using my Facebook profile photo and some other details that I didn't want appearing on comments. At that point I deleted the Stuff profile and created another one using a local login.
About the only good thing about using FB or Google credentials to log in is their security is very good and there is less likelihood of your username or password being stolen.