The discrete nature of both plots comes from having only categorical predictors in your models.
A martingale residual is the difference between the observed number of events and the expected number of events (based on covariate values) for an individual at an event or right-censoring time. See this page for an outline. Martingale residuals (and their related deviance residuals) shouldn't be thought of in the same way as residuals from least-squares regressions, given the complications due to censoring in survival models. Martingale residuals do have some uses, as in estimating the functional form for a continuous predictor. See this page. I don't think of them as something to check routinely.
The scaled Schoenfeld residuals in the second plot represent the differences between an individual's covariate value at an event time and the risk-weighted average of the covariate values among all those at risk at that time. Again, with only a limited number of covariate values for a categorical predictor, you will have banding like this: at any event time, there is only a small number of possible differences of an individual's value from the risk-averaged value. The smoothed curve illustrates how the apparent regression coefficient for the predictor changes over time. It thus provides information about how well the proportional hazards assumption (PH) is met; under PH with a time-constant regression coefficient (and associated hazard ratio), that smoothed curve should be horizontal. It's thus a good idea to check these plots routinely. See this page for an explanation.