Like all things, the law can be abused by the wealthy and powerful who can afford to bring cases. But defamation happens all the time, at all levels, whether on Twitter or in newspapers. It can be hugely financially damaging to a publication so journalism students are drilled pretty hard on it. The most common cases are stuff like misidentification - imagine a paper runs a story about a local man being convicted of a crime, but runs the wrong picture, a totally different guy. That’s defamatory.
What’s also key is that defamation can also be implied - you don’t have to outright say something, each case is judged on the basis of what readers will take from it. Sally Bercow’s tweet “Why is Lord X trending 
” doesn’t mean anything out of context, but as Twitter was abuzz with the rumour that Lord X was a paedophile, it defamed him by adding credence to the rumours that he was guilty.
Even things like restaurant reviews can be defamatory. Saying “Chez Dan’s clearly uses the cheapest prawns available for this dish” would damage the reputation of the restaurant, and they could prove you wrong - though it could be fine to say “Chez Dan’s prawn dish tastes like they use the cheapest ones available”, since opinions are okay.
Think my favourite cases are Face Magazine vs Jason Donovan - they alleged he was gay, which he denied. Now, it’s not defamatory to say someone is gay per se, but his denial meant they were implying he was a liar, and it’s defamatory to accuse someone of being dishonest.
Also, I remember a case, though I can’t find it now, of Cilla Black suing a magazine for calling her a cunt, which she lost, because it was judged that it was someone’s honestly held opinion that she was a cunt and it wasn’t defamatory to express that.
