Summary: This paper is concerned with singular curve and critical curve for the periodic doubly nonlinear Lane-Emden type equation \(\frac{ \partial u}{ \partial t} - \operatorname{div}(| \nabla u^m |^{p - 2} \nabla u^m) = a(x, t) u^q\). In 2010, under a convex assumption on the domain \(\Omega \), J. Wang et al. [Appl. Math. Comput. 216, No. 7, 1996–2009 (2010; Zbl 1425.35092)] considered a partial case of \(p - 1 < \frac{ q}{ m} < p - 1 + \frac{ p - 1}{ m N} \). While, there are no results about the cases of \(\frac{q}{ m} \leq p - 1\) and \(\frac{Q}{ m} \geq p - 1 + \frac{ p - 1}{ m N}\) before the present work. In this paper, we fill the gap for these two cases and give a total classification for the exponents. Furthermore, by constructing a special blow-up sequence, we remove the convex assumption on the domain \(\Omega\), not only for the partial case considered in [loc. cit.] but also for other remaining cases.