How do economists research economic trends? How can they predict a boom or crash in the market?

1- Lots of source digging 

2- Extra Reading

3- Using Mathematics.The2 words "Boom & Crash" from Eco point of view it a "Must Happen Scenario"  during the human economic history, It will happen & keep circling no matter how good/bad you are.

I agree with previous answer. I can only add "predict the unpredictable". Couple of years fell myth, in which we all believed, that property prices will still rise. Subprime mortgage crisis has shown that it is not.