… derived algebraic geometry … higher algebra …generalized scheme
…E-∞ scheme, locally representable structured (∞,1)-topos
Let $k$ be a commutative ring.
A derived Deligne-Mumford stack (over $k$) is a generalized scheme in the sense of locally affine $\mathcal{G}$-structured (infinity,1)-topos for $\mathcal{G} = \mathcal{G}_{et}(k)$ the étale geometry (for structured (infinity,1)-toposes).
A 1-localic derived Deligne-Mumford stack is an ordinary Deligne-Mumford stack. See there for more details.
spectral Deligne-Mumford stack is quasi-compact, quasi-separated E-∞ algebraic space? precisely if it admits a scallop decomposition.
Notice that for generalized schemes the étale geometry (for structured (infinity,1)-toposes) $\mathcal{G}_{et}(k)$ is not interchangeable with the Zariski geometry $\mathcal{G}_{et}(k)$. Instead $\mathcal{G}_{Zar}(k)$-generalized schemes are derived schemes.
In the context of E-infinity geometry (spectral Deligne-Mumford stacks):
Jacob Lurie, section 1 of Quasi-Coherent Sheaves and Tannaka Duality Theorems
Jacob Lurie, section 1 of Proper Morphisms, Completions, and the Grothendieck Existence Theorem