This　paper　is　concerned　with　the　stability　of　traveling　waves　for　a　Keller-Segel　model　with　singular　chemotactic　term　and　zero　chemo-attractant　diffusion,　where　the　model　and　the　waves　are　established　to　explain　the　propagation　of　bacteria　pulses　along　a　capillary　tube　observed　in　Adler＇s　experiment　[1].　By　applying　the　detailed　spectral　analysis,　Evans　function　method　and　special　transformations　and　combining　with　some　numerical　simulations,　in　some　range　of　the　parameters　all　the　waves　are　shown　to　be　spectrally　stable　in　some　exponentially　weighted　spaces,　and　in　other　range　of　parameters　all　the　waves　are　shown　to　be　unstable　in　any　exponentially　weighted　spaces.　The　local　well-posedness　of　the　classical　positive　solution　to　the　Cauchy　problem　of　the　model　is　also　obtained　by　applying　semigroup　argument　and　some　special　transformations.　(c)　2021　Elsevier　Inc.　All　rights　reserved.