Let's analyze each statement:

A) Double bond is shorter than a single bond: This statement is correct. A double bond involves the sharing of four electrons between two atoms, leading to a stronger attraction and a shorter bond length compared to a single bond (sharing of two electrons). For example, C-C bond length is ~1.54 Å, while C=C bond length is ~1.34 Å.

B) $\sigma$-bond is weaker than a $\pi$-bond: This statement is incorrect. A $\sigma$-bond is formed by the head-on (axial) overlap of atomic orbitals, resulting in a strong bond. A $\pi$-bond is formed by the sideways (lateral) overlap of p-orbitals, which is less effective and results in a weaker bond compared to a $\sigma$-bond. While a double bond (one $\sigma$ and one $\pi$ bond) is stronger than a single bond (one $\sigma$ bond), the individual $\pi$ component is weaker than the $\sigma$ component.

C) Double bond is stronger than a single bond: This statement is correct. A double bond consists of one $\sigma$-bond and one $\pi$-bond. A single bond consists of only one $\sigma$-bond. Since a double bond has more shared electrons and a $\pi$-bond in addition to a $\sigma$-bond, it requires more energy to break than a single bond.

D) Covalent bond is stronger than a hydrogen bond: This statement is correct. Covalent bonds are intramolecular forces that involve the sharing of electrons and are typically much stronger (bond energies ranging from 150-1100 kJ/mol) than intermolecular forces like hydrogen bonds (bond energies typically 10-40 kJ/mol). Hydrogen bonds are electrostatic attractions between a hydrogen atom covalently bonded to a highly electronegative atom (like O, N, F) and another electronegative atom.

Therefore, the incorrect statement is B.

Correct Answer: (B)