Quantum Anomalies for Bosons
The worldsheet Weyl anomaly in bosonic string theory is an example. More generally in any dimension you can have trace and Weyl anomalies that break scale or conformal invariance, even in systems with only bosons.
According to the theory of Symmetry Protected Topological (SPT) States: The underlying ultraviolet (UV) theory (at the cutoff/lattice scale) could be formed by fundamental bosons or fundamental fermions. For a $d+1$-dimensional gapped ground state that cannot be deformed to a trivial ground state under local unitary transformations when it is protected by global symmetry, we then have $d+1$-dimensional SPT states. The $d$-dimensional boundary of SPT states cannot be regularized and UV-complete in its own dimensions, unless the global symmetry is realized in an anomalous non-onsite manner. Accordingly, the anomalous global symmetry cannot be gauged, thus it is similar to the 't Hooft anomaly in $d$-dimensional spacetime -- the obstruction to gauge the global symmetry in an onsite manner.
In this sense, the conventional gauge anomalies (including 't Hooft anomaly) is actually the anomalies of realizing the global symmetries in the UV complete and in its own dimensions. And the "gauge" part of this "gauge" anomaly occurs (1) when you try to couple the anomalous global symmetry to a non-dynamical background gauge fields, or (2) to promote the anomalous global symmetry to a dynamical gauge theory.
This idea applies to both continuous symmetry or discrete finite symmetry, it applies to The systems formed by fundamental bosons are bosonic SPT states.
These are some examples of bosonic anomalies through SPT states:
Ref 1: Bosonic Anomalies in 1+1d PRB,
Ref 2: Anomalies and Cobordism of Oriented/non-Oriented in any dimensions, but non-spin manifold (thus non-fermionic) [arXiv only]
Ref 3: pure gauge and mixed gauge-gravitational anomalies in 0+1, 1+1, 2+1, 3+1, and any dimensions PRL,
Ref 4: Bosonic anomalies in any dimensions through extended Group Cohomology with an additional $SO(n)$ group PRB.
Some propose that classifying the SPT states are directly related to classifying the distinct anomalies of a group $G$, say related to an exact sequence:
$d$-dimensional gauge anomalies of gauge group G
$\to$ $d + 1$-dimensional SPT phases of symmetry group G
$\to$ 0.
Related phenomenon for bosonic anomalies:
Induced Fractional Quantum Numbers at domain walls: Jackiw-Rebbi and Goldstone-Wilczek (via bosonization/fermionization in 1+1d)
Degenerate Zero Modes at boundary/domain walls: Haldane chain., Kitaev chain, etc.
The only possible way Bosons admit anomalies in flat space (analogous to the case for fermions you have mentioned) is when they do not admit a covariant lagrangian. These are called by a special name, "Chiral Bosons". The usual Bose lagrangian can always be regulated to give an action free of anomalies. See section 8 of this paper. http://www.sciencedirect.com/science/article/pii/055032138490066X