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Zero-day vulnerabilities in closed-source software are typically found through a combination of methods:

1. Reverse Engineering: Analysts disassemble the software’s binary code to understand its structure and find flaws.

2. Fuzzing: Automated tools input unexpected or random data into the software to trigger errors that reveal vulnerabilities.

3. Static and Dynamic Analysis: Analyzing the software’s behavior and structure, either in a static state (without running it) or during execution, to identify weaknesses.

4. Security Research and Testing: Experts manually explore the software for weaknesses using their knowledge of common vulnerabilities and attack vectors.