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This communication presents the miniaturized low-profile second-order (SO) bandstop frequency selective surfaces (FSSs) for single- and dual-band resonances. The unit cell comprises two identical first-order (FO) elements in a back-to-back configuration with the metal grid in between. Two thin dielectric substrates separate the three metal layers. The proposed FO unit cell consists of intercell-connected spiral resonators (SRs) with capacitively coupled cross loop dipole. The SO filter response of the proposed FSS exhibits a wide stopband ($S_{21} < -10$ dB) bandwidth (BW) of 27.6%. The unit cell has the periodicity of $0.162\lambda _{0}$ , where $\lambda _{0}$ is the free space wavelength at the lower passband. The novelty of the proposed work is a miniaturized simple unit cell geometry, at the millimeter-waveband but not limited to it, with polarization insensitive and high angular stable filter response up to 60° with small variations in the BW. An unconventional four-unit cell element FSS is designed for dual-band SO stopband resonances with the BWs of 8.3% and 13.8%. Prototypes of both FSSs were fabricated and their measured results were found consistent with the simulated filter responses. [ABSTRACT FROM AUTHOR]