Professor Bersani explains: "The expansion of this abbreviation [ SPQR ] is "Senatus Populusque Romanus", a phrase which may be translated as "the Roman Estate" that is, the whole of the civic body and its institutional system. This formula is significant because first of all it proclaims the whole of the civic body, represented by the Senate and the people (i.e. the sovereignty of the republic is one but is two-headed, and in it the Senate and the people are indivisible). However on the other hand this formula also discloses the prominence of the Senate over the people, and then underlines the limitations which the latter suffered."
A. Apollonius Cordus has noted: "...the inscription on the temple of Saturn: Senatus populusque Romanus incendio consumptum restituit. Here we have a singular verb applied to what one would normally regard as a plural subject: the senate and the people. I wonder whether perhaps the senatus populusque is here regarded as a single entity, effectively the whole of the Roman state." 
P. Memmius Albucius adds:
1. In most of the inscriptions that I know, containing "senatus populusque Romanus", this expression is followed by a singular
- ex. 1 : Senatus / populusque / Romanus / clivom / Martis / pecunia publica / in planitiam / redigendum / curavit (CIL 06, 01270 - Rome)
- ex. 2 : Se[nat]us po[pu]lusq[ue Romanus] / c[olonia]e [O]sti[ensi]u[m mu]ro[s et port]as dedi[t] / M(arcus) [Tulliu]s C[icer]o c[o(n)s(ul) fecit carav]itqu[e] / P(ublius) Cl[odius] Pul[cher tr(ibunus) pl (ebis) co]nsumm[avit pro]b[avit] (..) (CIL 14, 04707, Ostia, Ita)
- ex. 3 : Genio deivi Iuli / parentis patriae / quem senatus / populusque / Romanus in / deorum numerum / rettulit (CIL 09, 02628 - Isernia, Samnium, Ita)
- ex. 4 : Publ]io Me[mmio Regula et] L(ucio) Fu[l]cinio Trione c[o(n)s (ulibus)] senatus populusque Iptuccitanorum [hospitium] fecit (...) (IRP Cadiz 00503, Baetica, Prado del Rey, Esp.)
2. When the plural is used, it seems to be in later inscriptions, during the late Empire, often written in african provinces and not in a very "classic" language :
- ex. Senatus p(opulus)q(ue) Lepcitanor(um) / C(aio) Macri f(ilio) C(ai) Annonis / n(epoti) Phelyssam ob colum/nas et superficie(m) et fo/rum stratum honoris / caussa [sic] decreverunt / Balitho [M]acri f(ilius) [C]o[m]/[modus](IRT 00615, Leptis magna, Trip., Libya).
3. It is interesting to remember that classical authors did not *always* use "SPQR" in a joined/pasted expression. For instance, Cicero has several times used a balanced formula like : "senatus decrevit populusque jussit", well making the difference between the two powers.