These all have their own uses, it depends on what content you have and how you want users to find content that determins which should be used. Using all of them will certianly be confusing, as all of these are variations on the same theme.
If voting or reviewing is important to your content, this is a great way to go. However it's also technically complex; if you go by % good/bad ratings, a single Good vote puts you at 100%, which would put you at the top of a post with 999 good votes and one bad.
Ratings are great for sorting out quality, which is important for commerce. I might not care what the most popular headphones on Amazon are, I want the best ones. If your content requires less buy in or varies less in quality (like news), rating probably isn't that helpful. Ratings are helpful on Stack Exchange because there's a lot of content and it varies greatly in quality.
This is popularity as a function of time; only recently popular items will show up in a "trending" sort, though how you weight time/popularity in the algorithm is up to you. These are generally very up to date, to the minute or second. Think "stock ticker."
Trending is vital for news; no one goes to CNN or the BBC and wants to see old news plastered on the front page. I've also seen "recently popular" used to describe this sorting method, trending is a bit informal and less clear as to what it means.
Note that lots of users have trouble understanding what "trending" actually means; many people find it hard to grasp what trending means on twitter. I would try to avoid the actual wording of "trending" even if you use the sort pattern.
This is the big one for highly social communities (and is often the same as "popular" on those communities) but the name clearly explains why it's popular; people are talking about it. Facebook and Twitter sort out content (by the Top Stories and Trending sections respectively). This can also be a fit for news sites that emphasize discussion.
A risk is that flamewars and contraversial topics are often emphasized in this sort method. That's not always the face you want to present to users.
This one is very clear, use this title instead of "Most Popular" if you want to differentiate it from Most Discussed ect. This is generally the same as "Most Popular"
This is really vauge; it can really mean any of the other sort methods I've covered, and it will be ambiguous if you include others. Generally this means "most views" or a combination of views and discussion, which is sort of what Stack Exchange does, though we use votes, not discussed.
This is a good title to use if you only have one sort method, and it's generally taken to mean "of all time". The Oatmeal updates once every couple weeks and so they have a Most Popular (of all time, best I can tell) to show the best of the best of their content. This can be ideal if your content isn't time sensitive. Note that none of The Oatmeal's Most Popular posts are that dated; it's a comic, not a news site.
What Users Think
Some of these terms can be confusing, much more so when they're used together. What's "Most Viewed" and "Most Discussed" is explicitly clear, so they're an effective (though technical) way to sort things out. Highest rating is clear (at least as clear as your rating system) but it doesn't always apply.
Most Popular and Trending are certainly easy to mix up; if you have just one sort method, most popular is probably fine. It's okay to be a little vague if there's only one sort method; people have a general idea of what popular means; lots of people are reading or talking about it. Start to mix that up with other sort methods that are kind of like popular and you'll start to have trouble.
Whether you need multiple sort methods depends on your content; highest rated and most popular are certainly different and can provide value if popular tends to be different from high quality/higly rated posts. Controversial topics are likely to get mixed ratings, but large amounts of views/discussion.
If you don't see the value in multiple sorting methods, your users probably see even less. Keep it simple unless it's clear to you or your users that more sorting methods are needed. If good posts are buried because recent posts get all the attention, consider adding rating sorts. If old content is emphasized too much, consider sorting by time or "trending" (recent popularity).