We present a theoretical investigation of the instability of self-trapped Frenkel excitons in one-dimensional (1D) microcrystallites composed of molecules for which the intermolecular interaction can be suitably described by the Lenard-Jones potential. Using the tight-binding approach, we have found that with decreasing microcrystallite size the self-trapped exciton (STE) becomes less dominant with respect to the free exciton due to the decrease in the self-trap depth. For the microcrystallite size below a certain value, the STE state practically disappears due to the widening of the trapping range over the whole microcrystallite. The characteristic feature of the 1D system is that, for the range of values used for the material parameters, the STE state has a minimum energy lower than the free exciton band bottom, regardless of the trapping range. Moreover the self-trapping barrier does not exist between the free exciton band and the STE level. From a similar calculation we have also found that these situations differ from those of two-dimensional (2D) and three-dimensional (3D) systems, and the trapping range in the 2D and 3D systems is always narrower than that in the 1D system. By comparison with experimental results, it is suggested that the STEs in both bulk aromatic crystals and microcrystallites can be described better by a 1D model.