We investigate the mechanical vibration transmission in two-dimensional infinite and finite phononic crystals (PCs). The infinite PC consists of a periodic structure formed by square plates connected to the center of each of their nearest neighbors through a tiny beam. Numerical simulations using finite elements show a wide full bandgap for frequencies between 27 kHz and 32 kHz, approximately. Acoustic resonant spectroscopy was used to measure the PC frequency spectra for the different vibrations, using a finite PC consisting of 8 by 8 cells, which was designed with the same configuration as the infinite one. Experimental results corroborate the existence of a full complete bandgap predicted by the numerical method. However, the width was significantly reduced due to the appearance of edge and corner states. The border states were obtained numerically using a supercell. The measured wave amplitudes and the simulated ones present a great similarity. Some states appear located at the corners of the finite PC demonstrating that zero-dimensional states can also appear in two-dimensional phononic crystals.