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Measurements of charge distributions of fragments in the spontaneous fission of $^{252}\mathrm{Cf}$ have been performed using a unique detector setup consisting of a typical grid-ionization chamber coupled with a $\mathrm{\ensuremath{\Delta}}E\ensuremath{-}E$ charged particle telescope. We find that the fragment mass dependency of the kinetic-energy-averaged width of the charge distribution shows a systematically decreasing trend with obvious fluctuations. The variation of the widths of the charge distribution with kinetic energy shows a pan-like shape. This is due to the large number of neutrons emitted at the high excitation energies and cold fragmentation at the low excitation energies. Deviation of the kinetic-energy-averaged most probable charge ${Z}_{p}$ from the unchanged charge distribution (UCD), $\mathrm{\ensuremath{\Delta}}Z$, as a function of the mass number of primary fragments, ${A}^{*}$, changes from negative for mass asymmetric fission to positive near the symmetric fissions. Concerning the kinetic energy dependence of ${Z}_{p}$ given primary mass number ${A}^{*}$, obvious increasing tendencies of ${Z}_{p}$ with increasing kinetic energy are observed.

The dependence of correlations of neutron multiplicity $\ensuremath{\nu}$ and $\ensuremath{\gamma}$-ray multiplicity ${M}_{\ensuremath{\gamma}}$ in spontaneous fission of $^{252}\mathrm{Cf}$ on fragment mass ${A}^{*}$ and total kinetic energy (TKE) have been investigated by employing the ratio of ${M}_{\ensuremath{\gamma}}/\ensuremath{\nu}$ and the form of ${M}_{\ensuremath{\gamma}}(\ensuremath{\nu})$. We show for the first time that ${M}_{\ensuremath{\gamma}}$ and $\ensuremath{\nu}$ have a complex correlation for heavy fragment masses, while there is a positive dependence of ${M}_{\ensuremath{\gamma}}$ for light fragment masses and for near-symmetric mass splits. The ratio ${M}_{\ensuremath{\gamma}}/\ensuremath{\nu}$ exhibits strong shell effects for neutron magic number $N=50$ and near doubly magic number shell closure at $Z=50$ and $N=82$. The $\ensuremath{\gamma}$-ray multiplicity ${M}_{\ensuremath{\gamma}}$ has a maximum for $\text{TKE}=165--170$ MeV. Above 170 MeV ${M}_{\ensuremath{\gamma}}$(TKE) is approximately linear, while it deviates significantly from a linear dependence at lower TKE. The correlation between the average neutron and $\ensuremath{\gamma}$-ray multiplicities can be partly reproduced by model calculations.