We present the results of a careful numerical analysis of the stability problem for stationary needle-crystal solutions of the symmetric model of dendritic solidification. The major outcome is that such needle crystals are linearly stable, at least on a time scale relevant to side-branching phenomena. Our study also indicate that the tip of needle-crystals is very sensitive to external noises, thus supporting the selective amplification mecanism advocated by Langer Barbieri and Barber in [BBL] on the basis of a WKB analysis.