This paper presents an indirect adaptive control scheme for linear continuous-time systems. The estimated plant model is controllable and then the adaptive scheme is free from singularities. Such singularities are avoided through a modification of the estimated plant parameter vector so that its associated Sylvester matrix is guaranteed to be nonsingular. That property is achieved by ensuring that the absolute value of its determinant does not lie below a positive threshold. An alternative modification scheme based on the achievement of a modified diagonally dominant Sylvester matrix of the parameter estimates is also proposed. This diagonal dominance is achieved through estimates modification as a way to guarantee the controllability of the modified estimated model when a controllability measure of the estimation model without modification fails. In both schemes, the use of a hysteresis switching function for the modification of the estimates is not required to ensure the controllability of the modified estimated model. Both schemes ensure that chattering due to switches associated with the modification is not present. The results are extended to the first-order case when the input is subject to saturation being modeled as a sigmoid function. In this case, a hysteresis-type switching law is used to implement the estimates modification.