Never attempt "Big Bang". It almost always blows in your face, since it's a high-risk, desperate measure when everything else has failed.
Divide and conquer: This works well ... if your world has only two sides. In real software, you have to conquer so many fronts at the same time, you can rarely afford to live in a black-white fantasy.
I guess I've been using something like "Strangling" for most of my career: Gradually morphing bad old code into shiny new code. Here is my recipe:
Start somewhere, it doesn't really matter where. Write a few unit tests to see how to the code really behaves. Find out how often it does what you think it does and how often it doesn't. Use your IDE to refactor the code so you can test it.
After the first day, make a guess whether you've started at the right place to take this monster apart. If so, go on. If not, find a new place and start over.
Advantages of this strategy: It works in small steps, so the risk can be kept in check and if something breaks, if has to be in the code you've been working on last week.
Disadvantage: It takes a whole lot of time and you will feel frustrated because often, progress will just seem so slow until the "knot" pops and suddenly, everything starts fall into place as if by magic.
与恶龙缠斗过久,自身亦成为恶龙;凝视深渊过久,深渊将回以凝视…