Cinque Terre Trails

Google is your friend.

http://www.cinqueterretrekking.com/cinque-terre-trails-status

http://www.parconazionale5terre.it/sentieri_parco.asp?id_lingue=2

http://savevernazza.com/traveladvisortrailupdates/

The trail that is the most likely to get closed, and parts of it still are, is trail 2, the low or tourist trail. Many, many other more breathtaking trails are open and stay open. Things are fluid though, literally, if there is a lot of rain then trails could get closed because of fear of landslides.

aj,

As mentioned on the websites provided in the first reply, the Sentiero Azzurro trails from Monterosso to Vernazza and Vernazza to Corniglia are currently open. The other two are closed and not expected to re-open this year. There's an extensive network of other trails in addition to the S.A., so there's no shortage of hiking opportunities in that area.

One other important point to note is that the status of the trails can change from time-to-time depending on the weather. I'd suggest checking the websites again just prior to your trip.