By recording the interference patterns of the speckle fields and the reference beam with the charge coupled device （CCD）, and by using the digital Fourier transform technique, the amplitude and the phase distribution of the speckle fields are experimentally extracted. We find that at the tangential points of the zero curves of the real and the imaginary parts, a new kind of speckle phase singularities may appear. Different from the conventional singularities at the zero crossings of the real and imaginary parts with the monotonically spiral change of phase, this new kind of singularities has the property that the phase undergoes an increase and then a decrease around the singular point with nearly a symmetric distribution. We introduce the concept of quasi twin phase vortices to explain the formation of the new kind of phase vortices. Based on a theoretical study of the longitudinal autocorrelation function of the speckle intensities, we experimentally observe the propagation of the phase vortices of speckles. It is found that in planes at different propagation distances but within a longitudinal correlation length, the real and the imaginary parts of the complex speckle field vary considerably, but the position of the phase vortices and the angle of the zero crossings of the real and imaginary parts remain unchanged.