One thing that can be confusing. Suppose we have some coordinates ##t,x,y,z##. And we wish to know if the coordinate x is spacelike, timelike, or null at some point P. To do so, we have to associate the coordinate x with some vector field, and evaluate the length of the vector field at some particular point P. The length of the vector at the particular point P can be classified as time-like, space-like, or null, depending on the sign of the length and the sign convention that one adopts.
However, the answer depends on whether we look at the contravariant vector filed ##\partial_x## = ##\frac{\partial} {\partial x}##, or the covariant vector field dx.
Most PF posters seems to be the convention to look at the vector field ##\partial_x## rather than the covector field dx. But it's clearer to talk about whether a vector field is time-like, space-like, or null at some point P rather than to talk about whether a coordinate is time-like, space-like, or null at some point P, as we need a map from a coordinate to a vector field in order to perform the classification. Of course it also depends on what point P we choose - for instance, in the Schwarzschild coordinates (t,r,##\theta##, ##\phi##), ##\partial_t## is well-known to be a time-like vector outside the event horizion, null at the horizon, and space-like inside the horizon.
Because the Schwarzschild metric is diagonal, dt has the same sign as ##\partial_t##. But this is not true in general. For metrics that are not diagonal, such as the Painleve metrics, the sign of ##\partial_t## and dt can be different.