Safety is the fundamental guarantee for the sustainable development of mining enterprises. As the safety evaluation of mines is a complex system engineering project, consistent and inconsistent, even hesitant evaluation information may be contained simultaneously. Linguistic neutrosophic numbers (LNNs), as the extensions of linguistic terms, are effective means to entirely and qualitatively convey such evaluation information with three independent linguistic membership functions. The aim of our work is to investigate several mean operators so that the safety evaluation issues of mines are addressed under linguistic neutrosophic environment. During the safety evaluation process of mines, many influence factors should be considered, and some of them may interact with each other. To this end, the Muirhead mean (MM) operators are adopted as they are powerful tools to deal with such situation. On the other hand, to diminish the impacts of irrational data provided by evaluators, the power average (PA) operators are under consideration. Thus, with the combination of MM and PA, the power MM operators and weighted power MM operators are proposed to aggregate linguistic neutrosophic information. Meanwhile, some key points and special cases are studied. The advantages of these operators are that not only the interrelations among any number of inputs can be reflected, but also the effects of unreasonable information can be reduced. Thereafter, a new linguistic neutrosophic ranking technique based on these operators is developed to evaluate the mine safety. Moreover, in-depth discussions are made to show the robust and flexible abilities of our method. Results manifest that the proposed method is successful in dealing with mine safety evaluation issues within linguistic neutrosophic circumstances.