  The recent start of the fourth observing run of the LIGO-Virgo-KAGRA (LVK)
collaboration has reopened the hunt for gravitational-wave (GW) signals, with
one compact-binary-coalescence (CBC) signal expected to be observed every few
days. Among the signals that could be detected for the first time there is the
stochastic gravitational-wave background (SGWB) from the superposition of
unresolvable GW signals that cannot be detected individually. In fact, multiple
SGWBs are likely to arise given the variety of sources, making it crucial to
identify the dominant components and assess their origin. However, most search
methods with ground-based detectors assume the presence of one SGWB component
at a time, which could lead to biased results in estimating its spectral shape
if multiple SGWBs exist. Therefore, a joint estimate of the components is
necessary. In this work, we adapt such an approach and analyse the data from
the first three LVK observing runs, searching for a multi-component isotropic
SGWB. We do not find evidence for any SGWB and establish upper limits on the
dimensionless energy parameter $\Omega_{\rm gw}(f)$ at 25 Hz for five different
power-law spectral indices, $\alpha = 0, \, 2/3,\, 2,\, 3,\, 4$, jointly. For
the spectral indices $\alpha = 2/3,\, 2, \, 4$, corresponding to astrophysical
SGWBs from CBCs, r-mode instabilities in young rotating neutron stars, and
magnetars, we draw further astrophysical implications by constraining the
ensemble parameters $K_{\rm CBC}, \, K_{\rm r-modes}, \, K_{\rm magnetars}$,
defined in the main text.
