Biodegrability of straight chain and branched chain detergents
According to Scott and Jones1:

... problems arise with branched alkyl chains, a side chain methyl group or a gem-dimethyl-branched chain cannot undergo β-oxidation by microorganisms and must be degraded by loss of one carbon atom at a time (α-oxidation).

* text by the authors, highlighting and hyperlinking by me.
A study by Whyte et al2 on the degradation of alkanes by Rhodococcus yielded similar findings:

After 28 days of growth at 5°C in MSM containing 0.1% diesel fuel, Rhodococcus sp. strain Q15 degraded almost all the n-alkanes (C10 to C21) and, to a lesser degree, many of the branched alkanes found in diesel fuel.

P.s: both of these articles are open access, and explain the biodegradation process quite well. You can refer to them directly if you need more detail or specific examples.
References:

Scott, Matthew J., and Malcolm N. Jones. “The Biodegradation of Surfactants in the Environment.” Biochimica et Biophysica Acta (BBA) - Biomembranes, vol. 1508, no. 1–2, 2000, pp. 235–51. doi:10.1016/S0304-4157(00)00013-7.
Whyte, L. G., et al. “Biodegradation of Variable-Chain-Length Alkanes at Low Temperatures by a Psychrotrophic Rhodococcus Sp.” Applied and Environmental Microbiology, vol. 64, no. 7, 1998, pp. 2578–84. PMID: 9647833