  We report on structural, magnetic, conductivity, and thermodynamic studies of
FeSe$_{0.5}$Te$_{0.5}$ single crystals grown by self-flux and Bridgman methods.
The samples were prepared from starting materials of different purity at
various temperatures and cooling rates. The lowest values of the susceptibility
in the normal state, the highest transition temperature $T_c$ of 14.5 K, and
the largest heat-capacity anomaly at $T_c$ were obtained for pure (oxygen-free)
samples. The critical current density $j_c$ of $8 \times 10^4$ A/cm$^2$ (at 2
K) achieved in pure samples is attributed to intrinsic inhomogeneity due to
disorder at the cation and anion sites. The impure samples show increased $j_c$
up to $2.3 \times 10^5$ A/cm$^2$ due to additional pinning centers of
Fe$_3$O$_4$. The upper critical field $H_{c2}$ of $\sim 500$ kOe is estimated
from the resistivity study in magnetic fields parallel to the \emph{c}-axis.
The anisotropy of the upper critical field $\gamma_{H_{c2}} =
H_{_{c2}}^{ab}/H_{_{c2}}^{c}$ reaches a value $\sim 6$ at $T\longrightarrow
T_c$. Extremely low values of the residual Sommerfeld coefficient for pure
samples indicate a high volume fraction of the superconducting phase (up to
97%). The electronic contribution to the specific heat in the superconducting
state is well described within a single-band BCS model with a temperature
dependent gap $\Delta_0 = 27(1)$ K. A broad cusp-like anomaly in the electronic
specific heat of samples with suppressed bulk superconductivity is ascribed to
a splitting of the ground state of the interstitial Fe$^{2+}$ ions. This
contribution is fully suppressed in the ordered state in samples with bulk
superconductivity.
