  A long-term measurement was conducted to search for $\alpha$, double-$\alpha$
and double-$\beta$ decays with $\gamma$ quanta emission in naturally occurring
osmium isotopes. This study took advantage of two ultra-low background HPGe
detectors and one ultra-low background BEGe detector at the Gran Sasso National
Laboratory (LNGS) of the INFN. Over almost 5 years of data were taken using
high-purity osmium samples of approximately 173 g. The half-life limits set for
a decays of $^{184}$Os to the first $2^+$ 103.6 keV excited level of $^{180}$W
($T_{1/2} > 9.3 \times 10^{15}$ yr) and of $^{186}$Os to the first $2^+$ 100.1
keV of $^{182}$W ($T_{1/2} > 4.8 \times 10^{17}$ yr) exceed substantially the
present theoretical predictions that are at level of $T_{1/2} \sim (0.6 - 3)
\times 10^{15}$ yr for $^{184}$Os and $T_{1/2} \sim (0.3 - 2) \times 10^{17}$
yr for $^{186}$Os. New half-life limits on the 2EC and EC$\beta^+$ decay of
$^{184}$Os to the ground and excited levels of $^{184}$W were set at level of
$T_{1/2} > 10^{16} - 10^{17}$ yr; a lower limit on the $2\beta^-$ decay of
$^{192}$Os to the $2^+$ 316.5 keV excited level of $^{192}$Pt was estimated as
$T_{1/2} > 6.1 \times 10^{20}$ yr. The half-life limits for $2\alpha$ decay of
$^{189}$Os and $^{192}$Os were set for the first time at level of $T_{1/2} >
10^{20}$ yr.
