I'm trying to use different distance metrics like Euclidean, Manhattan, cosine, chebyshev among other distance metrics in my k-means algorithm to calculate distances between the data points and the centers. In what situation would one distance metric be more useful over the other in a clustering scenario? [Comparing all the above mentioned distance metrics]
You ask a great question. Deza & Deza's Dictionary of Distances catalogs hundreds of families of distance functions. And there are others that are not mentioned there but in other books such as Cichocki et al Nonnegative Matrix and Tensor Factorizations
In summary, the choice of metric can certainly influence the results of even basic statistical data analysis tasks. It's a shame that statisticians understand very well that results are relative to such loss functionals, but applied scientists typically do not understand this and will compute and report an answer, as opposed to the answers.