Step 1: Understanding molecular vibrational modes.
For a molecule like CO$_2$, it has several vibrational modes, including stretching and bending vibrations.
Step 2: Analyzing CO$_2$ molecule.
A CO$_2$ molecule has three vibrational modes that are IR-active: two asymmetric stretching modes and one bending mode.
Step 3: Conclusion.
Thus, the correct number of vibrational modes detectable in IR spectroscopy for CO$_2$ is 3. The correct answer is (C) 3.