$ABCD$ is a quadrilateral on the complex plane whose four vertices satisfy $z^4+z^3+z^2+z+1=0$. Then $ABCD$ is a
$\textbf{(A)}~\text{Rectangle}$
$\textbf{(B)}~\text{Rhombus}$
$\textbf{(C)}~\text{Isosceles Trapezium}$
$\textbf{(D)}~\text{Square}$
1. We start with the given equation for the vertices of the quadrilateral:
   \[
   z^4 + z^3 + z^2 + z + 1 = 0
   \]
   This equation is known to have roots that are the non-unity fifth roots of unity. To see why, consider the factorization:
   \[
   (z-1)(z^4 + z^3 + z^2 + z + 1) = z^5 - 1
   \]
   This shows that the roots of \(z^4 + z^3 + z^2 + z + 1 = 0\) are the fifth roots of unity excluding \(z = 1\).

2. The fifth roots of unity are given by:
   \[
   z_k = e^{2\pi i k / 5} \quad \text{for} \quad k = 0, 1, 2, 3, 4
   \]
   These roots are evenly spaced on the unit circle in the complex plane. Excluding \(z = 1\) (which corresponds to \(k = 0\)), the remaining roots are:
   \[
   z_1 = e^{2\pi i / 5}, \quad z_2 = e^{4\pi i / 5}, \quad z_3 = e^{6\pi i / 5}, \quad z_4 = e^{8\pi i / 5}
   \]

3. To determine the shape formed by these points, we plot them on the complex plane. The points \(z_1, z_2, z_3, z_4\) form a regular polygon (specifically, a quadrilateral) inscribed in the unit circle.

4. We need to determine the type of quadrilateral formed by these points. Since the points are the vertices of a regular polygon inscribed in a circle, the quadrilateral is symmetric. However, we need to check if it is a specific type of quadrilateral such as a rectangle, rhombus, isosceles trapezium, or square.

5. By symmetry and the properties of the fifth roots of unity, the quadrilateral formed by \(z_1, z_2, z_3, z_4\) is an isosceles trapezium. This is because the opposite sides are equal in length, and the angles between adjacent sides are equal, but not all sides are equal in length (which would be required for a square or rhombus).

Conclusion:
The quadrilateral \(ABCD\) formed by the vertices \(z_1, z_2, z_3, z_4\) is an isosceles trapezium.

The final answer is \( \boxed{ \text{Isosceles Trapezium} } \)