This paper deals with maintenance strategies with minimal repairs for single-unit repairable systems which are subject to competing and dependent failures due to degradation and traumatic shocks. The main aims are to study different approaches for making a minimal repair decision (i.e. time-based or condition-based) which is a possible corrective maintenance action under the occurrence of shocks, and to show under a given situation which approach can lead to a greater saving in maintenance cost. Two age-based maintenance policies with age-based minimal repairs and degradation-based minimal repairs are modeled, and their performance is compared with a classical pure age-based replacement policy without minimal repairs. Numerical results show the cost saving of the maintenance policies and allow us to make some conclusions about their performance under different situations of system characteristic and maintenance costs. It is shown that carrying out minimal repairs is useful in many situations to improve the performance of maintenance operations. Moreover, the comparison of optimal maintenance costs incurred by both maintenance policies with minimal repairs allows us to justify the appropriate conditions of time-based minimal repair approach and condition-based minimal approach.