There were regulations, though most of the ones we know about date to the imperial era. The following builds (so to speak) upon an older answer to a similar question...

The Romans had no concept of zoning. Although certain districts were associated with, say, upper-class houses or warehouses, commercial and residential buildings were always jumbled, as were the houses of the rich and poor.

Building codes existed, though they seem to have been honored largely in the breach. We hear most about regulations limiting the height of new buildings. Augustus, for example, restricted new buildings on public streets (but not, of course, his own projects) to 70 Roman feet (Strabo 5.3.7), and even bored his dinner guests by reading them treatises on the dangers of excessively tall buildings (Suet., Aug. 89). Trajan apparently lowered the maximum to 60 feet. It was fairly easy, however, for buildings to circumvent these limits, since the parts of an insula (apartment block) that were not directly on a public street were apparently not subject to height restrictions. In the imperial era, one towering Roman insula was so tall that it became a tourist attraction.

Roman buildings were also supposed to have narrow spaces between them (e.g. Dig. 8.2.14) to allow light into the rooms, but it is clear that quite a few insulae were bounded by party walls on three sides, and that light would have been very limited in many apartments.

After the great fire of 64 CE, Nero briefly attempted to rebuild Rome on a more regular model, "with broad thoroughfares, buildings of restricted height, and open spaces, while colonnades were added as a [fire] protection to the front of the tenement-blocks" (Tac., Ann. 15.43). This initiative, however, seems to have been short-lived, since imperial Rome continued to be characterized by twisting lanes and jerry-built structures. An frequently-quoted passage from Juvenal's Third Satire, written decades after Nero's reforms, laments:

"we inhabit a city supported for the most part by slender props, for that is how the landlord holds up the tottering house, patches up gaping cracks in the old wall, bidding the inmates sleep at ease under a roof ready to tumble about their ears..." (193-6)

Later in the Empire, finally, we hear more and more about regulations forbidding the scavenging of building materials from existing structures.

As mentioned earlier, almost all of the known regulations date to the imperial era, when Augustus and his successors transformed central Rome into a museum and theater of their power. During the Republican era, when there was no authority with remotely comparable means or opportunity to transform the cityscape, there appear to have been only a few laws concerned with buildings. None survive; but to judge from the evidence of municipal charters, it seems to have been forbidden to tear down buildings without authorization, and it may have been illegal (as it would be under Augustus and his successors) to construct apartments taller than a set height.