I have 3 non tantrumming kids ages 5, 3 and 2. Fits were instantly nipped. You need to be calm and methodical with spanking at the very beginning of the fit before it escalates. Calm warning, firm consequence if it continues for one second. Dont' let it get out of control, or you'll have to just wait for the next one. You want to teach him to decide against it in the BEGINNING. He's 3, so he's probably been throwing fits for over a year, and his maturity and stubbornness have set in. Therefore, it may take lots of repetition (but maybe not) rather than just one or two attempts like it would have when he was younger, and he may get worse at first to try to intimidate you. Be calm, be consistent, be clear what his consequence is for. He'll quit.
Clarification: This will NOT crush his spirit or remove his emotions. He will still get mad, sad, scared, and he'll be able to express those emotions better. You'll know if he does melt down that he's tired or it's a legitimate situation. (Don't discipline when over tired or hungry). A younger sibling is NOT an excuse. Mine all had new babies to deal with, and my third was a natural rager, but even she controls her fits at 2.
Example, the other week at the pool, she wanted to play with some cars another kid brought while the kid wasn't around. When I told her to come back to the baby pool, she said "No!", and stomped her foot. I gave her the "very last warning for defiance and saying no to momma calm final warning". Instead of coming to where I was, she started the gutteral "I'm gonna let a tantrum rip" growl (thinking that because we were at a brand new public place she might get away with it) and turned red and her eyes started welling with tears. I said, "OK then" calmly with a step toward her to take her to the restroom for a swat-which she knew from experience-to which she said, "No! I want splashy." and was laughing and splashing in the pool 2 seconds later feeling proud she outsmarted me, completely forgetting she was mad. This is a child born with the temper form h_ll and she would definitely be one of those kids screaming and kicking and choking and head banging all the time if she was allowed.
Here are the actions that will make his tantrums much worse:
Ignoring. NEVER ignore a fit if you want them to stop.
Time outs. They're just places to throw fits.
Toy removal. Who cares. There' still a house full of toys.
Talking and saying you wish he wouldn't do it. It's just attention.
These methods are ways to "put up with fits" while waiting for the kids to outgrow them on their own. BUT, you can add them in after 3 good swats for 3 years old, like, remove a favorite toy for a day AFTER his real consequence, or remove him from a fun activity for a time out AFTER his consequence. It sounds extremely harsh, but you wont' have to do it many times for it to click, and you're not ANGRY when you do this, just matter of fact and firm in response to his choice of behavior.
If you're tough about this, you'll still be able to cuddle your lovey boy. You'll get to enjoy him and cuddle him more when he quits the constant fits. My 3 year old son is attached to me at the hip and a constant cuddler. He's only been spanked once since December and we use no time outs or anything else. Kids love firm boundaries within a fun loving home. All my kids are sweet and happy with their emotions firmly in tact and hardly ever need discipline. You want to prevent the behavior, not constantly discipline it.
Good luck! This method is very successful at having happy, respectful kids: Back to Basics Discipline by Janet Campbell Matson.