Consider the following subscheme $X$ of $\mathbb A^2_\mathbb R$ which is made by a curve minus a point and "plus" an isolated point $p$:
Which is the simplest way to show that $X$ is singular at the isolated point $p$? Maybe I can say that in $p$ the tangent space is $\mathbb A^2_\mathbb R$... is this correct?
You should be a little more careful; the tangent space to $X$ at the isolated point is not $\mathbb{A}_\mathbb{R}^2$, but is in fact the trivial vector space.
However, this is fine! Essentially, a scheme is smooth if the tangent space has constant dimension at each point. In this case, your tangent spaces have dimension either zero or one, depending on where you are in your scheme, and so it can't be smooth.
In general, if you have a scheme consisting of several components not all of which are the same dimension, then it is not smooth. It could, however, be locally smooth which you do have in this case.