This article investigates the optical soliton solutions of the time-fractional Biswas–Arshed (BA) equation, which is a new model for soliton transmission through optical fibers where the eigen-phase modulation is negligible and thus removed. We have studied both models including M-truncated or beta derivative operators. Firstly, the time-fractional BA equations have been transformed into a nonlinear ordinary differential equation using appropriate wave transformations. The singular, singular-periodic, and dark soliton solutions have been derived using the extended rational sine–cosine and sinh–cosh techniques have been applied and the existing conditions for the obtained solutions have been introduced. Utilizing a computer algebraic system, we have verified that all of the derived solutions satisfy the fractional Biswas–Arshed equation. Additionally, we have compared the results for the beta and M-truncated derivatives and have examined how the equation’s parameters affect the amplitude of the solitons in 2D and 3D graphical demonstrations. The attained solutions may aid in the comprehension of wave propagation in optical fibers and may contribute to the telecommunication sector.