What is the shape of a molecule whose central atom is $sp^3$ hybridised and has no lone pairs (e.g. $\mathrm{CH_4}$)?

Four $sp^3$ hybrid orbitals point to the vertices of a regular tetrahedron, giving a bond angle of $109.5°$. With no lone pairs to compress these angles, the molecular geometry equals the electron-pair geometry. Methane ($\mathrm{CH_4}$) is the textbook example.