Who has a right to be pissed off at whom?

Muslim demonstrators may say so (picture).
But actually blasphemy is not allowed in Denmark. When muslim demonstrators then wonder why the Danish government does not crack down on Jyllandsposten and its editor, they must try to understand that this is not the way the system of government works in Denmark.
Denmark is a constitutional monarchy. There is a Queen, but she has only ceremonial functions. She is not to have any political influence or power whatsoever. Real power is vested in the Danish parliament (Folketinget). Members of parliament are elected by all voters, i.e. all citizens above 18 years of age. The whole system is based on the division of powers. Legislative power resides in parliament. Executive power resides in the government. And the judicial power resides in the courts. The parliament passes the laws. The government and the ministries execute the laws through administration, and the courts make judgments based on the rules in the laws.
Freedom of speech is guaranteed in the Danish constitution and Danish law generally. Therefore the Danish government cannot “bring Jyllandsposten to justice” for publishing the cartoons, nor stop the paper printing them. This can only be done if the drawings violate Danish legislation. Perhaps they do. On this point, however, there is some doubt. The issue may have to be tried in court.
Freedom of speech – even in the Danish variant of this basic human right - is not a right that entitles a person to automatically say anything imaginable about a person or a subject matter. Freedom of speech is limited by other laws, for instance the penal code’s rules about libel. Nor are people allowed to talk derisively or condescendingly about religious groupings. According to the penal code article 140:

§ 140(Danish penal code). Anybody who publicly mocks or derides the dogma of any religious community in this country or its divine worship is to be punished with a fine or imprisonment up to 4 months. .

By dogma may be meant not only the Qu'ran's text, but perhaps also religious rules defined by the muslim clergy. There is no ban against depicting the Prophet (pbuh) in the Qu'ran. It is on the other hand part of what may be called dogma. It is possible to find pictures of the Prophet (pbuh), also pictures made by muslim artists. In these instances the purpose, however, is not to mock. § 140 has only led to a court sentence once, in 1938 in an anti-semitic trial. There was an attempt at using it again in the beginning of the 1970's when a singer made a song making some mild mockery of Jesus. It did not lead to a sentence. The question of whether Jyllandsposten may be sentenced, will depend on the court's assessment of Jyllandsposten's intent. If the intent was mockery of religion, there is a clear case for a sentence. Jyllandsposten has claimed that it wanted to test freedom of expression after some artists claimed they were exerting self-censorship because of fear of muslim retribution after the Van Gogh murder in the Netherlands. So it all depends on the judges' assessment of the validity of this explanation.

There is also an article in the penal code that bans expressions of a racist kind:

§ 266B (Danish penal code): Anybody who publicly or with deliberation makes utterances in a broader circle, by means of which a group of people are threatened, mocked or degraded because of their race, skin colour, national or ethnic background, faith or sexual orientation, is to be punished with a fine or by imprisonment for up to 2 years.

11 muslim organisations in Denmark have reported the publication of the drawings to the police. A district attorney in the Danish town of Viborg has refused the case on grounds that the violation of § 140 is not clear enough for the public prosecutor to take action. This decision has been appealed to the General Attorney in Copenhagen, who is considering the case. His decision will be final. In case of a confirmation of the district attorney’s decision, the minister of Justice in the Danish government can decide that the case be tried in court.
This is the only procedure available for the prosecution and eventual punishment of Jyllandsposten and/or the cartoonists. It may seem meagre to the demonstrators who are enraged at the treatment of their prophet. But there is no other way, unless the Danish constitution and system of government be changed. And that is not an option. It's difficult to change the Danish constitution. It must be done by a majority in parliament. Then again by a newly elected parliament, and finally by referendum in which a majority that constitutes at least 40 per cent of voters vote for the constitutional amendments.