The catenary is a curve that appears frequently in the nature, describes the shape that a flexible cable takes when it hangs free tied at its ends. It is used in the construction of suspension bridges, churches, buildings, electric cable laying, etc. It also appears in the human body in some parts such as the dental arch. On the other hand, in recent years Fractional Calculus (FC) has positioned itself as an important tool to describe phenomena more approximately than conventional calculus. In this work new families of curves based on the catenary and different definitions of fractional derivatives are proposed. Starting from a known ordinary differential equation of second order. Then, Atangana-Baleanu–Caputo (ABC) and Conformable fractional derivative are applied, as well, an alternative expression based on the Caputo-Dzhrbashy (CD) is proposed. With the (CD) and (ABC) fractional derivatives, the fractional derivatives depend on the Mittag–Leffler (ML) function and the fractional order \(0<\alpha \le 1\) . Finally, some numerical simulations are shown, and is demonstrate that, if \(\alpha =1\) the solution converge to the original catenary curve.