Lisca has completed the determination of the smooth concordance orders of 2-bridge knots.
Adam Levine has shown the following knots have infinite smooth concordance order, building on the work of Grigsby, Ruberman, and Strle:
9_30, 9_33, 9_44, 10_58, 10_60, 10_102, 10_119, 10_135, 11a_4, 11a_8, 11a_11, 11a_24, 11a_26, 11a_30, 11a_52, 11a_56, 11a_67, 11a_76, 11a_80, 11a_88, 11a_126, 11a_160, 11a_167, 11a_170, 11a_189, 11a_233, 11a_249, 11a_257, 11a_265, 11a_270, 11a_272, 11a_287, 11a_288, 11a_289, 11a_300, 11a_303, 11a_315, 11a_350, 11n_12, 11n_48, 11n_53, 11n_55, 11n_110, 11n_114, 11n_130, 11n_165,
E. Grigsby, D. Ruberman, S. Strle, Knot concordance and Heegaard Floer homology invariants in branched covers, math.GT/0701460.
S. Jabuka and S. Naik, "Order in the concordance group and Heegaard Floer homology," math.GT/0611023.
A. Levine, "On knots of infinite smooth concordance order," arxiv.org/abs/0805.2410.
P. Lisca, "Sums of lens spaces bounding rational balls," arXiv:0705.1950.
C. Livingston and S. Naik, "Knot concordance and torsion," Asian J. Math. 5 (2001), no. 1, 161--167.
C. Livingston and S. Naik, "Obstructing four-torsion in the classical knot concordance group," J. Differential Geom. 51 (1999), no. 1, 1--12.
A. Tamulis, " Knots of ten or fewer crossings of algebraic order 2," J. Knot Theory Ramifications 11 (2002), no. 2, 211--222.