Summary: ``In this article, the following Kirchhoff-type fractional Laplacian problem with singular and critical nonlinearities is studied: $$ \cases a + b\Vert u\Vert^{2\mu -2} (-\Delta)^s u = \lambda l(x)u^{2^*_s -1} + h(x)u ^{-\gamma}, &\text{in }\Omega\\ u > 0, &\text{in }\Omega,\\ u = 0, &\text {in }\Bbb R^N\backslash \Omega, \endcases $$ where $s\in (0, 1)$, $N > 2s$, $(-\Delta)^s$ is the fractional Laplace operator, $2^*_s = 2N/(N - 2s)$ is the critical Sobolev exponent, $\Omega\subset\Bbb R^N$ is a smooth bounded domain, $l\in L^ \infty(\Omega)$ is a non-negative function and $\max \{l(x), 0\} \nequiv 0$, $h\in L^{\frac{ 2^*_s}{ 2^*_s +\gamma-1}} (\Omega)$ is positive almost everywhere in $\Omega$, $\gamma\in (0, 1)$, $a > 0$, $b > 0$, $\mu\in [1, 2^*_s /2)$ and parameter $\lambda$ is a positive constant. Here we utilize a special method to recover the lack of compactness due to the appearance of the critical exponent. By imposing appropriate constraint on $\lambda$, we obtain two positive solutions to the above problem based on the Ekeland variational principle and Nehari manifold technique