We need a jurisdiction, because burglary is a statutory crime. Since this is a residence, in Washington, under RCW 9A.52.025(1), the trespasser might have committed residential burglary:
A person is guilty of residential burglary if, with intent to commit a
crime against a person or property therein, the person enters or
remains unlawfully in a dwelling other than a vehicle.
While other states may have a provision requiring intent to commit a felony, or intent to commit a crime other than trespassing, the Washington statute does not say "felony" or "other than trespassing". A person can enter unlawfully by accident, or with no intention to be there unlawfully, in which case the person committed the act without the requisite criminal intent. If a person intends to enter but not remain, and they enter, they have committed the misdemeanor of trespassing. The question of whether unlawful entering with the intent to unlawfully remain satisfies the "intent to commit a crime" called for in the Washington statute has not been directly addressed in case law, as far as I know.
While the state generally shoulders the burden of proving beyond reasonable doubt the elements of the crime, in Washington, under RCW 9A.52.040, the accused may bear the burden of disproving intent (this is a permissive inference on the fact-finder's part):
In any prosecution for burglary, any person who enters or remains
unlawfully in a building may be inferred to have acted with intent to
commit a crime against a person or property therein, unless such
entering or remaining shall be explained by evidence satisfactory to
the trier of fact to have been made without such criminal intent.
This does not mean that the state shoulders no burden of proof regarding intent, according to State v. Newton:
A jury may, however, infer the defendant's specific criminal
intent from his or her conduct if it is not "'patently
equivocal'" and instead 'plainly indicates such intent as a matter of logical probability'
In this instance, the burglary conviction was overturned because
no evidence shows his intent was anything other than to show
her she could walk
The relevant fact is that the accused was under the influence of PCP, and believed that God had told him that his disabled mother could walk, and he broke into the residence to tell her this.
Pursuant to State v. Bergeron where the conviction for attempted burglary was upheld, the court held that
The intent to commit a specific named crime inside the burglarized
premises is not an "element" of the crime of burglary in the State of
The intent required by our burglary statutes is simply the intent to
commit any crime against a person or property inside the burglarized
The court observed in this case that "there is absolutely no evidence in the record to prove what specific crime it was the defendant intended to commit inside".
What distinguishes Bergeron from Newton (the latter, not precedential, citing the former) is that there was a good case that Bergeron had an intent to commit some crime, though not a specific identifiable crime; but for Newton, there was no evidence that defendant intended to commit any crime whatsoever, even remaining unlawfully. In the hypothetical instance, the fact that the criminal smashed a vase is evidence of an intent to commit a crime (destruction of property). It's unlikely that a mere assertion "I just did that on the spur of the moment" would carry any weight. If the circumstances make it more likely that he entered intending to do something wrong, such as vandalism, the elements of burglary are present/
If the defendant claims some innocent reason for trespassing, such as escaping a riot, he may avail himself of a defense (RCW 9a.52.090) that
The actor reasonably believed that the owner of the premises, or other
person empowered to license access thereto, would have licensed him or
her to enter or remain
If there is no evidence of a riot, such a defense is not credible. Alternatively, if the house is for sale, the "I thought it would be okay" defense is more credible.