An ADE singularity is a orbifold fixed point locally of the form $\mathbb{C}^2/\Gamma$ with $\Gamma \hookrightarrow SU(2)$ a finite subgroup of the special unitary group given by the ADE classification (and $SU(2)$ is understood with its defining linear action on the complex vector space $\mathbb{C}^2$).
These singularities have crepant resolutions, obtained by repeatedly blowing up at singular points. The resulting exceptional fiber (the blow-up of the singular point, an ALE space) is a union of Riemann spheres that touch each other such as to form the shape of the corresponding Dynkin diagram.
(graphics grabbed from Wijnholt 14, part III)
Peter Slodowy, Simple singularities and simple algebraic groups, in Lecture Notes in Mathematics 815, Springer, Berlin, 1980.
Miles Reid, The Du Val Singularities $A_n$, $D_n$, $E_6$, $E_7$, $E_8$ (pdf)
Kyler Siegel, section 6 of The Ubiquity of the ADE classification in Nature , 2014 (pdf)
MathOverflow, Resolving ADE singularities by blowing up
Families of examples of G2 orbifolds with ADE singularities are constructed in
Ashoke Sen, A Note on Enhanced Gauge Symmetries in M- and String Theory, JHEP 9709:001,1997 (arXiv:hep-th/9707123)
Luis Ibáñez, Angel Uranga, section 6.3.3 of String Theory and Particle Physics: An Introduction to String Phenomenology, Cambridge University Press 2012
Martin Wijnholt, slides 6-10 of String compactification, PITP 2014 (pdf)
See also at F-branes -- table