The statement regarding conformations of ethane that is incorrect is:

Option C - Eclipsed conformation is the most stable conformation.

To explain why Option C is incorrect, let's analyze all the options:

Option A: "The dihedral angle in staggered conformation is $$60^{\circ}$$." This statement is incorrect. In staggered conformation, the dihedral angle, or the angle between the planes containing the hydrogen atoms, is actually $$60^{\circ}$$. However, the staggered form is not unique; it refers to the form where any given pair of hydrogens on adjacent carbons has this dihedral angle, but rotating the molecule can yield a variety of staggered conformations with different specific hydrogen atoms at the $$60^{\circ}$$ dihedral angle. The full rotation of one ethyl group relative to the other through $$360^{\circ}$$ creates a cycle where the staggered conformation occurs every $$120^{\circ}$$.

Option B: "Ethane has infinite number of conformations." This statement is correct. The rotation around the carbon-carbon single bond (sigma bond) is relatively free, so ethane can theoretically adopt an infinite number of conformations between the two extremes of staggered and eclipsed. These conformations can have very slight differences in angles, resulting in finely varied overlap of electron clouds.

Option C: "Eclipsed conformation is the most stable conformation." This statement is incorrect, which makes it the answer to the question. The eclipsed conformation is actually the least stable conformation of ethane due to the increased torsional strain from electron repulsion. The electron clouds of the bonding pairs of hydrogens on adjacent carbons are aligned in such a way as to be as close to each other as possible, increasing repulsion. The most stable conformation is the staggered conformation, where the groups are as far apart as possible, reducing repulsion.

Option D: "The conformations of ethane are inter-convertible to one-another." This statement is correct. As you rotate around the carbon-carbon single bond, the conformations continuously change from one to another. This interconversion can be quite rapid and occurs without the need for a high energy input under normal conditions.

Thus, the incorrect statement is Option C.