The paper is concerned with the nonlinear Dirac equation \[ D_g\psi = f(x)|\psi|_g^{\frac{2}{m-1}}\psi \quad \text{ on the sphere }(S^m,g),\quad m\geq 2,\tag{\(\ast\)} \] where \(D_g\) is the Atiyah-Singer-Dirac operator. Due to the critical exponent this equation is a spinorial analogue of the Yamabe equation. The authors deal with two special cases: the case where \(g=g_{S^m}\) is the standard round metric and \(f:S^m\to\mathbb{R}\) is close to \(1\); and the case where \(f\equiv1\) and \(g\) is close to \(g_{S^m}\). In the first case they consider functions of the form \(f=1+\varepsilon H\), and show that \((*)\) has a solution for \(\varepsilon\) small, for generic \(H\) in a certain class of functions \(S^m\to\mathbb{R}\). In the second case they construct metrics on \(S^m\) that are close to \(g_{S^m}\) but not conformally equivalent such that \((*)\) has multiple solutions. As a consequence of the first result they obtain multiple conformal embeddings of \((S^2,g_{S^2})\) into \((\mathbb{R}^3,g_{\mathbb{R}^3})\) whose images have mean curvature \(1+\varepsilon H\). As a consequence of the second result they can prove the strict inequality in the spinorial analogue of Aubin’s inequality from [B. Ammann et al., Math. Z. 260, No. 1, 127–151 (2008; Zbl 1145.53039)]. The existence results for solutions of \((*)\) are based on perturbation methods in critical point theory.