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I am wondering something about the breaking of conformal symmetry: I know that it can be broken at the quantum level, anomalously, but I never encountered or heard about a model where it is broken "à la Higgs" with a potential whose true minimum would spoil this invariance (e.g. making appear a particular energy scale). I guess we would then get some Goldstone bosons, would there be something special about them?

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It's completely common that a conformal symmetry is spontaneously broken, i.e. in a Higgs-like way. For example, a conformal field theory may have scalar fields and if their expectation value is $v$, they define a preferred mass scale, breaking the scaling symmetry. This field itself is, however, the Goldstone mode for the scaling as well because changing the vev in a spacetime-dependent way is equivalent to making a conformal transformation etc. This attribution is a bit misleading because one could only talk about Goldstones for internal symmetry and conformal symmetry or scaling isn't one. – Luboš Motl Sep 25 '12 at 9:08

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