One question that divides libertarians is the legitimacy (or lack thereof) of privileges such as copyright, patent, and trademark -- those usually subsumed under the rubric of "intellectual property".
Many libertarians, particularly those unaware of online politics, don't have a strong opinion on the topic. A good number consider it a minor matter in the light of greater government violations of ordinary everyday rights, including regular property.
Some natural-law libertarians believe in a right of authors and inventors to control others' copying of their creations. They usually believe this right should have all the conventional attributes of property, including perpetual inheritance. They differentiate between the intellectual property (e.g. a blueprint, or music) and its physical manifestation (e.g. the machine or a copy of the music), the former being used to create the latter. They believe, therefore, that an owner's control over the use of his/her property extends to control over the use of intellectual property. Nonetheless, such a right (in their view) would exist regardless of whether government chose to enforce it.
Other natural-law libertarians believe that intellectual property is but a monopoly privilege that wouldn't exist but for government intervention, and that it should be abolished. To them, whatever secrecy and exclusivity are to exist should be achieved out of voluntary contracts, the cost of which are to be born by those who try to achieve secrecy and exclusivity.
A few minarchists, including most Objectivists, accept the mainstream justifications of copyright et al. as monopolies useful to the market, regarding them as necessary acts of government to promote industrial and authorial innovation. Other libertarians consider the grant of monopoly to be beyond the pale of minarchism.
Many libertarians consider copyright and patent to be forms of enclosure -- illegitimate government creation of exclusive privilege by prohibiting most individuals from accessing commons. Copyright and patents are government-granted monopolies on production, and no better than a government-granted monopoly on producing food or Internet service.
Trademark, unlike copyright and patent, can be construed as a protection against fraud and misrepresentation: it ensures that others cannot abuse a successful product's good name to promote an inferior knockoff. Since most libertarians believe that fraud should be criminal, they agree in this regard with trademark law.
However, in many jurisdictions the concept of trademark dilution has developed to protect trademarks as a property right, securing the investment the trademark owner has made in establishing and promoting a strong mark without regard to liklihood for confusion. This has even been used to limit free speech about a product, something few libertarians would be likely to defend.
Anarcho-capitalists might differ as to the final answer, but agree that the optimal answer would emerge from a free market in justice and protection.