At what smallest $n$ is there a convex $n$-gon for which the sines of all angles are equal and the lengths of all sides are different?
1. **Understanding the problem**: We need to find the smallest \( n \) such that there exists a convex \( n \)-gon where the sines of all angles are equal and the lengths of all sides are different.

2. **Analyzing the angles**: If the sines of all angles are equal, then each angle must be either \( x^\circ \) or \( (180^\circ - x)^\circ \) for some constant \( x \). This is because \(\sin(x) = \sin(180^\circ - x)\).

3. **Checking \( n = 3 \)**:
   - For a triangle, the sum of the interior angles is \( 180^\circ \).
   - If all angles are \( x^\circ \), then \( 3x = 180^\circ \implies x = 60^\circ \).
   - This implies the triangle is equilateral, which contradicts the requirement that all sides must be different.
   - Therefore, \( n = 3 \) is not possible.

4. **Checking \( n = 4 \)**:
   - For a quadrilateral, the sum of the interior angles is \( 360^\circ \).
   - If all angles are \( x^\circ \), then \( 4x = 360^\circ \implies x = 90^\circ \).
   - This implies the quadrilateral is a rectangle, which cannot have all sides different.
   - If the angles are \( x^\circ, x^\circ, (180^\circ - x)^\circ, (180^\circ - x)^\circ \), then the quadrilateral is an isosceles trapezoid or a parallelogram, which also cannot have all sides different.
   - Therefore, \( n = 4 \) is not possible.

5. **Checking \( n = 5 \)**:
   - For a pentagon, the sum of the interior angles is \( 540^\circ \).
   - Consider a pentagon with angles \( 120^\circ, 120^\circ, 120^\circ, 120^\circ, 60^\circ \).
   - The sines of these angles are all equal: \(\sin(120^\circ) = \sin(60^\circ) = \frac{\sqrt{3}}{2}\).
   - We can construct a pentagon with side lengths \( 1, 2, 4, 3, 6 \) respectively, ensuring all sides are different.

6. **Verification**:
   - The given pentagon with side lengths \( 1, 2, 4, 3, 6 \) and angles \( 120^\circ, 120^\circ, 120^\circ, 120^\circ, 60^\circ \) satisfies the conditions.
   - The sines of all angles are equal, and all side lengths are different.

Conclusion:
\[
\boxed{5}
\]