Is the Laplacian a vector or a scalar?

The "Laplacian" is an operator that can operate on both scalar fields and vector fields. The operator on a scalar can be written,

$$\nabla^2 \{\} = \nabla \cdot (\nabla \{\})$$

which will produce another scalar field.

The operator on a vector can be expressed as

$$\nabla^2 \{\} = \nabla (\nabla \cdot \{\})\,\,-\nabla \times (\nabla \times \{\})$$

which will produce another vector field.

In Cartesian coordinates, both operators can be written

$$\nabla^2 \{\} = \frac{\partial^2 \{\}}{\partial x^2}+\frac{\partial^2 \{\}}{\partial y^2}+\frac{\partial^2 \{\}}{\partial z^2}$$

where it is evident that operation on a scalar (vector) field transforms into a scalar (vector) field.