R.B.
At this age, her behavior sounds perfectly normal--unfortunately for us mommas!
She wants her voice to be heard. She's trying to assert her independance and decision-making skills. Kind of like if you pick our her clothes for her and she starts a tantrum because she doesn't want to wear that particular outfit (even though it's one of her favorites). She's screaming solely because she didn't get to make the decision on something in her little 3 1/2-year old world.
As these little wonders get older, they do/should get to make more decisions about the world they interact with. But these decisions should not be open-ended like "what do you want to do today?" Those options are endless and can make the decision overwhelming for a youngin'.
You may find that if you start to give her her own choices and decisions to make, she'll respond better at those times when she doesn't have a choice. And you can even control the outcome to be something you want, too. The choice should never be between more than 2 things. So, for your example about the bike trailer, next time you could ask "Do you want to bring your doll or toy car with you in the trailer?" Now you've given her a choice between two reasonable toys to take with in the trailer, but you've taken the decision about whether or not she's actually IN the trailer away without her even knowing it!
You'll still likely get some good tantrums because that's just how kids are and we, as the parents, do the best we can to manage and corral the tantrums. But I've had more success with the method I've described than I've had with yelling back at my kids when they start massive tantrums!
Good luck to you all! This, too, shall pass :)