We consider different fractional Neumann Laplacians of order s ∈ (0, 1) {s\in(0,1)} on domains Ω ω n {\Omega\subset\mathbb{R}^{n}}, namely, the restricted Neumann Laplacian (- Δ Ω N) R s {{(-\Delta-{\Omega}^{N})^{s}-{\mathrm{R}}}}, the semirestricted Neumann Laplacian (- Δ Ω N) Sr s {{(-\Delta-{\Omega}^{N})^{s}-{\mathrm{Sr}}}} and the spectral Neumann Laplacian (- Δ Ω N) Sp s {{(-\Delta-{\Omega}^{N})^{s}-{\mathrm{Sp}}}}. In particular, we are interested in the attainability of Sobolev constants for these operators when Ω is a half-space.