The complicated nature of singularities associated with topological transition in the plane Taylor-bubble problem is briefly discussed in the context of estimating the speed of the fastest smooth Taylor-bubble in the absence of surface tension. Previous numerical studies were able to show the presence of a stagnation point at the tip of the bubbles for dimensionless speed F < 0.357 but were incomplete in characterizing the topology of these bubbles at the tip for values of F > 0.29 due to difficulties in obtaining numerical solutions with well-rounded profiles at the apex. These difficulties raise the question whether the bubbles rising at a speed F β (0.29, 0.357) are smooth, pointed or spurious. This issue has led us to carefully scrutinize certain asymptotic behavior of the Fourier spectrums of the numerical solutions for a wide range of values of F and to extend these results in an appropriate limiting sense. Our findings indicate that these plane bubbles with F < 0.35784 (accurate up to four decimal places) have well-rounded profiles at the apex. The purpose of this paper is to describe our approach and its use in arriving at the above conclusion.