It probably very much depends on what is it that you want to achieve.
I can imagine that specific profiles of people receive preference in given situations due to their sexual orientation or even gender.
A guy I knew once insisted that he felt he was given an au pair job because he was a str8 male. The wife did not want to challenge the fidelity of her husband
I also believe that being str8 or at least being perceived as such helps if you are building a career in a homophobic environment. Globally, there is no shortage of such environments.
Straight privilege can be encountered with housing issues, too. A few landlords are hung on the notion that being gay is all about wild parties, and disturbing the neighboring residents. Hence, the housing preference (even though this may be illegal) goes to the str8 as opposed to the gay guy.
By and large, the str8 privilege is being relegated to the realm of urban legends in the vast parts of the world these days. Quite a few employers are actually giving preference the gay guys since they see family as a possible source of worker's absence. A few landlords have also understood that some gay men happen to be affluent, since they are not spending on children, and are more likely to pay the rent on time than some of the str8 guys.
Many str8 business leaders have understood that discrimination based on sexual orientation is harmful to their business interests beyond the negative publicity it may create.
The playing field is being levelled.