What things that people write on Twitter should be sackable offences?

I don't have any absolutes on this where tweeting X should = a sacking. But I do think the following should be taken very seriously and be part of disciplinary proceedings that may lead to sackings:

Death threats, including Twitter-acceptable ones like Kill All TERFS

Incitements to violence

Discriminatory tweets based on characteristics protected by the Equalities Act*

Online bullying.

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*This one is hugely open to interpretation.

For me it would be tweets that show a level of disdain and/or prejudice to certain groups, rather than discussions about delicate topics that involved one or more people in this groups...

For example:
I'd class persistently misgendering somebody, even after they've told you their name and their gender identity, would be worthy of disciplinary action. So would tweets that portray trans people as in any way "less" than 'cis' people...

But I don't think that following an account like FairPlayForWomen or engaging in discussions about the Gender Reform Act or possible issues with self-identification should be. Even if these might be reported to employers as transphobia.
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Wherever your line is, employers need to investigate thoroughly and realise how easy it is for someone to create fake tweets and put in vexatious claims of malpractice.
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For example, I do have concerns about Priyamvada Gopal's views. She comes across on here as being racist, incendiary and intolerant which I don't think is good for someone who works for a university...

Her tweet about resisting the urge to kneecap white men caused a stir and some people use that, alongside her 'white lives don't matter' tweet to pressurise Cambridge University into sacking her...

But while the tweet looks bad in isolation, in the context of the thread it's in and an interview that occured two days before, the tweet makes perfect sense and it's clear the point she is making... https://www.independent.co.uk/arts-entertainment/films/news/liam-neeson-rape-black-man-attack-cosh-cold-pursuit-sexual-assault-interview-a8760866.html

I find the frenzy on here to get people sacked to be mad, and a little bit worrying.

But at the same time, if your Twitter account is linked to your real name and your employer, you have to accept that your tweets can and will reflect on your employer.

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If you are the type of person who will report a person you don't like for tweeting [Tweet X], then you should apply the same principles you do like, otherwise you're just a bully.
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And employers should resist the kneejerk, quick sacking reactions we've seen on the past two days with the nurse who used the phrase "all lives matter" and the woman who create the TikTok view about stabbing people.
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Instead, take a breath, resist the demands of the mob, investigate thoroughly and take the action that you see as proportionate and fair.

Sacking someone just to appease a baying Internet mob is such a short-sighted move.

And if you're going to write anything remotely controversial on the Internet, don't link your account to your employer in anyway because sadacts on here will use it against you.

My last school's social media policy listed swearing on social media as a disciplinary offence.