In the present research, vibration and instability analysis of a viscoelastic Y-shaped single-walled carbon nanotube conveying fluid is carried out. The surrounding viscoelastic medium is simulated by various models such as Kelvin-Voigt, Maxwell, standard linear solid Reissner, and nonlocal models. The size effects are considered based on modified couple stress theory. In order to achieve more accurate results, fourth-order beam theory is utilized. Surface stress effects are considered based on Gurtin-Murdoch theory. In addition, effects of the asymmetry of Y-shaped single-walled carbon nanotube are also taken into account. Regarding fluid-structure interaction, the equations of motion as well as boundary conditions are derived using Hamilton's principle and solved by means of hybrid analytical-numerical method. Regarding the temperature changes on visco-Pasternak foundation, the effects of different surrounding medium models are discussed in detail. The overall results indicated that the stability and vibration characteristics of Y-shaped single-walled carbon nanotube conveying fluid are strongly dependent on damping coefficient. The results of this work are hoped to be useful in design and manufacturing of nanodevices where Y-shaped nanotubes act as a basic element.