In New York, there are two types of rent regulation, Rent Stabilization (RS) and Rent Control (RC). As a general matter, Rent Control is a better deal, but to be in RC the tenant (or his/her predecessor) had to be in possession since 1978. Once the RC tenant moves out, the apartment (again as a general matter) reverts to Rent Stabilization. It is nearly impossible to succeed to RC apartment unless you are a close family member living with the tenant of record.
Most apartments in the City either are or were Rent Stabilized because the buildings are either old enough to qualify or the owners accepted certain tax benefits which required them to apply RS. An apartment may fall out of RS for several reasons, most of which resulted from some strange legislative compromises. First, if a building goes coop or condo, in most instances a rent regulated tenant may remain in the building under regulation, but when that tenant vacates, the apartment becomes unregulated. Second, if the legal regulated rent of an apartment exceeds $2,000 per month, it will become unregulated when the existing tenant vacates. Third, if the regulated rent of an apartment exceeds $2,000 per month and the tenants have a household income in excess of $175,000 per year, the landlord can apply to deregulate the apartment. Needless to say, there are lots of twists, turns and quirks to all of this.
That being said, if an apartment is Rent Stabilized, anyone who rents it is entitled to the benefits of RS. Although it is becoming less common to find RS apartments in the lower half of Manhattan because of the $2,000 vacancy decontrol provisions, they are widely available in upper Manhattan and the outer boroughs. Under RS, your rent will only increase by a percentage set by the Rent Guidelines Board each year, though there are some factors that can augment the increase. One significant restriction of RS is that the apartment must be your primary residence. Also there significant limitations on subletting, including that you can sublet a RS apartment for a maximum of two years.
The succession rules under RS are arcane, but generally one has to have been a close family member (or living in a family-like relationship) with the tenant of record for at least two years before the tenant of record vacates. In that case, you can succeed to the apartment under the existing RS lease, but be prepared for a legal fight to prove that you actually qualify for succession unless your case is particularly clear-cut.
So, the easiest way to get a RS apartment is just find one and rent it. It will be difficult to do in Manhattan, and you may have to pay a broker to get one, but it should be fairly easy to do in the outer boroughs, unless you are looking in one of the hot areas. The thing is that in the outer boroughs, the RS apartments are often at about the same rent level as the unregulated apartments, so it doesn’t benefit you all that much, unless your area gets hot, in which case it can help keep rent increases in check.