Is the Higgs a quantum field or a particle? The Higgs is not detected in the asymptotic data, so it is possible that there is no particle interpretation for the Higgs quantum field. Indeed, the Higgs potential is only positive definite if the quartic term is included --- the quadratic term corresponds to a negative mass term. It would seem, therefore, that the Higgs field does not have an on-shell mass spectrum, so that there is no straightforward particle interpretation.
One can say that there is an effective field theory in which there is a resonance near a given mass that we will call the Higgs resonance, but in the absence of a pure mass shell spectrum (that is, if there is a continuous mass spectrum), it is generally taken in QFT that there is no particle interpretation. The resonance is clearly not a $\delta$-function, so is there some other precise way in which we can call the Higgs a particle?
Of course this doesn't call into question the empirical effectiveness of the Standard Model of Particle Physics, it only asks about its interpretation and about how we put the Mathematics into words.
I was somewhat struck by Rolf Heuer's observation (this morning) that this is the first observation of a scalar particle. Indeed, according to the SM, there are no quantum fields that have non-zero mass terms in the absence of interactions. In the absence of interactions, the Higgs field is a massless scalar field. Should we say that it is the Higgs interaction that gives mass to the standard model? (EDIT: Is it better to say that every term that is not quadratic in the fields contributes towards the effective masses of each of the asymptotically observed fields? Or what alternative phrasing is closer to the Mathematics of the interacting fields?)
EDIT(2, $\scriptstyle\mathsf{see\ below\ for\ the\ comment\ that\ prompted\ this\ possible\ rephrasing}$): Is there any part of the definition of "particle" that is not a matter of convention? Does the Higgs cross that bar?
 A: 
The Higgs is not detected in the asymptotic data, so it is possible that there is no particle interpretation for the Higgs quantum field. 

Possible but not plausible. The particle interpretation comes out of quantum
field theory quite naturally, so if you want to get rid of a particle but keep
the field, it seems that you will have to replace much of QFT as well. Not an
easy task. Note that there are other fields whose particles have not been
detected asymptotically --- the top quark is but one example --- so the Higgs
is not special in this regard.

Indeed, the Higgs potential is only positive definite if the quartic term is included --- the quadratic term corresponds to a negative mass term. It would seem, therefore, that the Higgs field does not have an on-shell mass spectrum, so that there is no straightforward particle interpretation.

This is a common misconception. To have a theory with a well-defined
particle-spectrum, one should consider perturbations near a stable vacuum. At
this vaccuum, it is not necessary that scalar fields like the Higgs have a zero
expectation value. In the case of the Higgs, at zero expectation value we do
not have a stable vacuum, so it is meaningless to discuss the mass spectrum
there, as you say. But the correct thing to do is to consider the field near
the value where its potential is at a minimum, namely to write $\phi = \phi_0 +
\delta\phi$, where the potential $V$ is minimized at $\phi_0$, and $\delta\phi$
is the dynamical field. The field $\delta\phi$ does have a positive
mass-squared, and this is what we call the mass of the Higgs particle.
Note that shifting the field is also the realistic thing to do, because $\phi$ really does
have a non-zero expectation value in nature.

One can say that there is an effective field theory in which there is a resonance near a given mass that we will call the Higgs resonance, but in the absence of a pure mass shell spectrum (that is, if there is a continuous mass spectrum), it is generally taken in QFT that there is no particle interpretation. The resonance is clearly not a δ-function, so is there some other precise way in which we can call the Higgs a particle?

The fact that the resonance has a width (and is not a $\delta$-function) is true for all particles that can
decay. It is a question of semantics whether you would like to call such
resonances particles, and physicists have decided that it makes sense to do so.
But again note that the Higgs is not unique in having a width: the top, $W$,
$Z$, and other fundamental particles also have this property. Would you like to stop calling them particles as well?

I was somewhat struck by Rolf Heuer's observation (this morning) that this is the first observation of a scalar particle. 

It is the first observation of a fundamental scalar particle. Pions have been
observed a long time ago.

Indeed, according to the SM, there are no quantum fields that have non-zero mass terms in the absence of interactions. In the absence of interactions, the Higgs field is a massless scalar field. Should we say that it is the Higgs interaction that gives mass to the standard model? 

Whether to say that it is the Higgs field or its interactions that give mass to
other particles is a question of semantics. Certainly both are necessary.

Is there any part of the definition of "particle" that is not a matter of convention? Does the Higgs cross that bar?

Particles are the perturbative excitations of quantum fields. Once you expand
around the correct vacuum, as I explained above, this definition applies to the
Higgs particle as well.
A: What is the difference between a field described by $E=E_{0}\sin(\omega t)$ and a field described by $\phi = \phi_{0}+ \phi_{1}\sin(\omega t)$?  The particle is the excitation of the field, not the field itself.  The only difference between the Higgs and anything else is that its vibrations are relative to a nonzero background, rather than vacuum.  There's nothing in Maxwell's equation that prevents you from quantizing around non-zero backgrounds, btw.
