In the paper, the authors concisely review some closed formulas and applications of special values of the Bell polynomials of the second kind for some special sequences and elementary functions, explicitly present closed formulas for those sequences investigated in [F. T. Howard, A special class of Bell polynomials, Math. Comp. 35 (1980), no. 151, 977–989; Available online at https://doi.org/10.2307/2006208], and newly establish some closed formulas for some special values of the Bell polynomials of the second kind
