It's horrible isn't it? ds1 is 22m and has tantrummed for aaages, but was getting out of hand at 19m (around the time of ds2's arrival). He's been better in the last few weeks, but still has the capacity to scream himself silly. One of my problems is I suppose because the tantrums have been so awful, I lost the capacity to tell what was a "normal" tantrum, what was only to be expected in a child of his age, and started feeling that he was just a defiant, stubborn "naughty" boy .
I've asked family, searched MN and read books, and kind of think you have to find what works well for your lo, but here's some things that seem to have helped in the last couple of months.
When a tantrum is in the offing, I try very hard to distract ds1 to prevent it. So I know triggers - time to leave a place he's having fun, having to share his tractors, taking things he's got hold of (because bad mummy hasn't put them out of reach!) away from him - I try and find ways to take his attention elsewhere. For example, pointing out something through the window ("oooh look, ds1, there's a bird/plane/lorry" in a humiliating Mary Poppins type voice), or starting to sing a silly song, or tickling him.
When the tantrum avoiding hasn't worked, I second the ignoring it thing. He can scream for 20-30mins even when ignored, so it is horrible and I just try and busy myself so it doesn't upset me so much. Sometimes I will say, once, "stop screaming, ds1, I don't talk to you when you're screaming". Then when tantrum over, I tell him he's good for stopping screaming, hug him etc but relatively swiftly move on - toddlers don't hold grudges and it's important that once it's over, it's over.
One tip was to, instead of saying "no" or "don't do that" (which I sometimes feel I spend all day doing!) is to say "we don't hit people" etc, because toddlers like to feel included, like to feel part of a group.
When he is being good, I tend to overdo the positivity, I think to partly compensate for how much I feel I've been crap with him dealing with his behaviour after ds2's birth. So if he's playing happily I will tell him how good he is, how much I love him, give him cuddles etc. I'm really trying hard to praise the good behaviour, though it can be hard - if he plays quietly I don't notice him so much (esp if busy with ds2) so forget to praise him, it's when he's "naughty" he draws attention to him, so it's very much been a conscious effort to do this. The theory is that he learns that good behaviour gets attention, bad behaviour doesn't.
And the mantra "it's a phase, it's a phase" is v helpful, though damn it's a long phase! It's just helpful to remind yourself that he is still your lovely, gorgeous son, he's just going through what all kids go through - it's just you notice it a lot more when it's your child drumming their heels on the floor!
Also think his behaviour has been better with the nicer weather - definitely think burning the energy off him (which I couldn't do so well when heavily pg/post cs) has made him happier.
Sorry, v long post - I've been thinking about this issue for a very long time! I hope it helps, even if it's just a matter of knowing you're not alone!