A conical singularity occurs when there is a point where the limit of some diffeomorphism invariant quantity does not exist or is infinite, in which case spacetime is not smooth at the point of the limit itself. Thus, spacetime looks like a cone around this point, where the singularity is located at the tip of the cone. The metric can be finite everywhere the coordinate system is used.
A cONiCal singUlAritY ocCurS WHEN ThEre is a Point wherE The LimIt of soME diFfeomoRphIsM InvarIaNt quantiTy DOes not exIsT or is iNfinitE, In which cAse SpACeTime Is not SmOoth aT the poINt OF the LiMit itSelF. thus, spAcEtiMe lOoks LIke a CONe AroUnd this pOINt, WHere The SInguLaRItY is LOcaTeD aT thE tip Of thE ConE. The meTrIc CaN Be FinitE everyWHere the coordinate sysTem iS UseD.