I packaged a few of the private ones with this weekend project: http://sharedcount.com Unfortunately, only Facebook does URL normalization for you, so dumb things like trailing slashes have a big effect on the count.
edit 2: graphs.facebook.com will aggressively and unfortunately normalize any YouTube link to youtube.com. See, for example, http://graph.facebook.com/http://www.youtube.com/watch?v=ZB8...
This give the number of shares for youtube.com, not this individual video. I am not sure how to get facebook share stats for a particular YouTube video.
LinkedIn: http://www.linkedin.com/cws/share-count?url=http://news.ycom...
These APIs are marked private (as opposed to Facebook's) so use at your own risk.