Step 1: Recall the regulator's role.
The Securities and Exchange Board of India, known as SEBI, was set up to oversee and regulate the securities market, including stock exchanges.
Step 2: Check what the other bodies do.
CBI is India's main investigative agency for crime, CID handles state level criminal investigations, and SHCIL is a depository services company, not a market regulator.
Step 3: Confirm the match.
None of CBI, CID, or SHCIL has the legal authority to regulate stock exchanges, while SEBI was created specifically for this purpose.
Final Answer:
SEBI regulates the stock exchanges in India. \[ \boxed{\text{SEBI}} \]