In this paper we study the tangent space to the Hilbert scheme
H
i
l
b
d
P
3
\mathrm {Hilb}^d \mathbf {P}^3
, motivated by Haiman’s work on
H
i
l
b
d
P
2
\mathrm {Hilb}^d \mathbf {P}^2
and by a long-standing conjecture of Briançon and Iarrobino [J. Algebra 55 (1978), pp. 536–544] on the most singular point in
H
i
l
b
d
P
n
\mathrm {Hilb}^d \mathbf {P}^n
. For points parametrizing monomial subschemes, we consider a decomposition of the tangent space into six distinguished subspaces, and show that a fat point exhibits an extremal behavior in this respect. This decomposition is also used to characterize smooth monomial points on the Hilbert scheme. We prove the first Briançon-Iarrobino conjecture up to a factor of
4
3
\frac {4}{3}
, and improve the known asymptotic bound on the dimension of
H
i
l
b
d
P
3
\mathrm {Hilb}^d \mathbf {P}^3
. Furthermore, we construct infinitely many counterexamples to the second Briançon-Iarrobino conjecture, and we also settle a weaker conjecture of Sturmfels in the negative.