A simple question about the scattering amplitude $\mathcal{M}$ in QFT Every scattering amplitude that I see have all the tensor indices contracted but spinor indices floating around, whose only disappear after you square the amplitude and do the sum and average over spins. 
My question relates to my lack of knowledge about spinors: is the amplitude $\mathcal{M}$ always a scalar? Even though not having the spinor indices summed over?
 A: Yes: the scattering amplitude is a Lorentz scalar.
Spinor indices are not floating around. They are contracted too. They are not shown explicitly to unclutter the notation. In general, all suppressed indices are contracted. The standard convention is that one suppresses as many indices as possible as long as  the expression is unambiguous. Only vector indices are sometimes explicitly shown, but mostly for historical reasons. One could suppress them too.
In this sense, an expression of the form
$$
\bar u\gamma v\ \bar v\gamma u
$$
is a short-hand notation for
$$
\bar u\gamma^\mu v\ \bar v\gamma_\mu u
$$
which is itself a short-hand notation for
$$
\bar u_a\gamma^\mu_{ab} v_b\ \bar v_c\gamma_{\mu cd} u_d
$$
which is itself a short-hand notation for
$$
\sum_{abcd\mu}\bar u_a\gamma^\mu_{ab} v_b\ \bar v_c\gamma_{\mu cd} u_d
$$
etc.
It should be clear that the first expression is preferred: it carries all the relevant information, as concisely as possible.
If you want to play around with spinor indices in order to gain some confidence, check out Srednicki, Part II (Spin One Half). Read as many chapters as possible/necessary. You can find a free copy on his webpage.
A: All Lorentz indices are summed over, but it is more subtle than you might think. The amplitude for particles of helicity/spin greater than zero is not a scalar. The reason for this is that the external polarizations/spinors transform under the little group (Lorentz transformations that leave the momenta invariant). 
For instance, gluon scattering amplitudes in four dimensions transform under the $U(1)$ little group of massless particles as 
$$
\mathcal{M} \rightarrow e^{-2i\theta  \sum_i h_i}\mathcal{M}\,,
$$
where $h_i$ is the helicity of each particle. For more details take a look at chapter 2.5 of Weinberg's QFT I, or to section 2.6 of this review for a more modern perspective.
