This paper is devoted to new facts about the so-called unit-Rayleigh distribution. More precisely, we contribute by: (i) proving new stochastic ordering results, (ii) bringing new characteristics on the form analysis of its main probabilistic and reliability functions with an emphasis on the mode(s), (iii) exposing manageable expressions of the incomplete and probability weighted moments, (iv) investigating distributional properties of the order statistics, (v) giving the expression of the reliability coefficient, (vi) providing a tractable expression for the Tsallis entropy and (vii) proposing some bivariate unit-Rayleigh distributions. On a practical level, two data sets are analyzed and adjusted, revealing that the unit-Rayleigh distribution can be a better alternative to standard one-parameter unit distributions, such as the one-parameter Kumaraswamy, Topp-Leone, one-parameter beta, power and transmuted distributions.