The authors deal with the control system \[ \dot x(t)=f(x(t),u(t)), \] where \(f:M\times N\to TM\) is a smooth function with manifolds \(M,\;N\) of dimensions \(m\) and \(n\) respectively. If \(M_0\) and \(M_1\) are two subsets of \(M\), then \(\mathcal {U}\) is the set of admissible controls such that the associated trajectories steer the system form an initial point of \(M_0\) to a final point in \(M_1\). For such controls \(u\) the cost function is defined by \[ C(t_f,u)=\int_0^{t_f}f_0(x(t),u(t))\,dt, \] where \(f_0:M\times N\to R\) is smooth. Applying the Pontryagin maximum principle one receives the Hamiltonian system for the trajectory \(x(.)\) connected with an optimal control \(u\in \mathcal{U}\), the adjoint vector \(p(.)\) and a nonpositive number \(p^0\) satisfying \((p(.),p^0)\neq (0,0)\). An extremal \((x(.),p(.),p^0,u(.))\) is called normal if \(p^0<0\) and abnormal if \(p^0=0.\) The authors present algorithms to compute the first conjugate time along a smooth extremal curve, where the trajectory ceases to be optimal. The article contains a review of second order optimality conditions. The computations are related to a test of positivity of the intrinsic second order derivative or a test of singularity of the extremal flow. Their algorithm is applied to the minimal time problem and to the attitude control problem of a rigid spacecraft. The algorithm involves both normal and abnormal cases.