Good question!  I can give you some ideas about the red flags when you know the definite need to check for extraneous solutions.

• When the denominator contains the unknown variable and you have to clear the denominator by multiplying the LCM of the denominators.  This operation has a possibility to introduce solutions to make the denominator $0$.
• When you have to square both sides to clear the radicals.  This operation makes it possible to introduce solutions to make one side positive and the other side negative on the original equation, but the new equation (after squaring) hold.
• If you divide the number line into intervals and solve with casework (such as equations involving absolute values), check the solutions in each case against the case itself to see if the solution falls in the range.  If not, that's an extraneous one.
• Anything with domain restrictions: denominators cannot be 0, things inside radicals with even index cannot be negative, can't take logarithms of negative numbers, base of logarithms cannot be negative or 1, etc.  As long as those are present in the equation, a red flag should be raised and you always check against those requirements.

The following should also be considered about losing solutions.

• Do not cancel any expression with unknown variables from both sides if they could be zero.  Those are solutions, and if you cancel them, you lose those solutions.
• If a quadratic equation has a parameter in the coefficient of $x^2$, then that is a red flag, because that leading coefficient may be $0$ and if that is $0$, you get a linear equation instead of quadratic.  The analysis of a linear equation is very different from a quadratic one, so you can miss some solutions that way.