Orientable circle bundle with torsion Euler class have been studied systematically. There are exactly the flat $SO(2)$-bundles, see "A Remark on Torsion Euler Classes of Circle Bundles" by Miyoshi or "Flat circle bundles, pullbacks, and the circle made discrete" by Oprea-TanrĂ©(I could not find a link for the latter).

It is a standard fact that any flat $G$-bundle over a (connected) finite cell complex $X$ can be written as $(\tilde X\times G)/\pi_1(X)$ where $\tilde X$ is the universal cover and $\pi_1(X)$ acts by deck transformations on the first factor, and via some homomorphism $\pi_1(X)\to G$ on the second factor. Thus all examples look like the one given by Anton.

As a caution I wish to point out that many people also studied flat circle bundles with $G=Diff(S^1)$. The answer there is different, namely one gets the so called Milnor-Wood inequality as a condition on the Euler class.

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Orientable circle bundle with torsion Euler class have been studied systematically. There are exactly the flat $SO(2)$-bundles, see "A Remark on Torsion Euler Classes of Circle Bundles" by Miyoshi or "Flat circle bundles, pullbacks, and the circle made discrete" by Oprea-TanrĂ© (I could not find a link for the latter).

It is a standard fact that any flat $G$-bundle over a (connected) finite cell complex $X$ can be written $(\tilde X\times G)/\pi_1(X)$ where $\tilde X$ is the universal cover and $\pi_1(X)$ acts by deck transformations on the first factor, and via some homomorphism $\pi_1(X)\to G$ on the second factor. Thus all examples look like the one given by Anton.

As a caution I wish to point out that many people also studied flat circle bundles with $G=Diff(S^1)$. The answer there is different, namely one gets the so called Milnor-Wood inequality as a condition on the Euler class.