How to check integrity of a file in Linux

Question

I want to verify whether a file has been modified or corrupted on my Linux system. I know that hash functions and checksums are commonly used, but I need clarification on:

• Which hashing tools (sha256sum, md5sum, shasum) are best for file integrity checks?
• How to compare a file’s current hash with a previously stored hash?
• How to use tripwire or aide for continuous integrity monitoring?

A step-by-step example of checking a file’s integrity and detecting unauthorized changes would be helpful.