As pointed out in the comments, you are asking about two different things: testing for a structural break in each of the parameters and estimating (or dating) the structural breaks, if any.
As for testing: The Nyblom-Hansen test would be a natural candidate here. It's a test based on the scores (i.e., contributions to the gradient of the model) and was first proposed by Nyblom (1989, Journal of the American Statistical Association, 84, 223-230) for a single parameter, later extended by Hansen (1992, Journal of Policy Modeling, 14, 517-533), and shown to by trend-resistant by Ploberger and Krämer (1996, Journal of Econometrics, 70, 175-185). Hansen (1992) proposes to apply this test jointly to all parameters (in your case: intercept and slope) but also separately to each parameter. As I have pointed out in Zeileis (2005, Econometric Reviews, 24, 445-466) the latter is fine but you should apply a simple Bonferroni correction to account for carrying out two separate tests (one for each parameter). This would give you what you want: separate tests for intercept and slope, either of which can or cannot be significant.
As for dating: I wouldn't know of an algorithm that estimates separate breakpoints in the two parameters. Usually, breakpoints are selected such that all parameters in the model change simultaneously, e.g., as in Bai & Perron (2003, Journal of Applied Econometrics, 18, 1-22). If you apply that method and get multiple breakpoints, you can also judge easily whether each change is driven more by the intercept or the slope. An idea to formalize separate breakpoint estimates would be an EM-type approach, i.e., estimate the breaks in the intercept given the breaks in the slope and vice versa and then iterate until convergence. But as I said above, I haven't seen something like this in the literature, yet.