I was just looking through a list of vetoes cast at the UNSC (http://research.un.org/en/docs/sc/quick/veto) and noticed several prevailing trends. One of these is the preponderance of vetoes cast by the USSR against applications of new members across continents, often at the same session/day, and some more isolated in time-frame. What were the circumstances that led to these vetoes, and why the USSR cast them so often? How different were these circumstances from each other, or were they quite similar of logic?
Just looking at some of the transcripts of sessions involving new membership, I cannot discern why some states were accepted while others proposed at the same time were not.
It was mainly to do with China. The United Nations was originally the name for the World War II Allies; if you read the UN Charter it clearly envisages something more like NATO than the international talking-shop that we now have. The Republic of China (ROC) was a major WW2 ally, and therefore had a seat on the Security Council. In 1946-49 there was a civil war in China, at the end of which the ROC government only controlled Taiwan, a few other islands, and China's seat at the UN. Most of the old China was controlled by the People's Republic of China (PRC). Mongolia, which had been part of imperial China, was partly controlled by the PRC (Inner Mongolia) and partly a separate puppet state of the USSR (outer Mongolia). The USSR wanted the PRC to get the Security Council seat and Mongolia to get UN membership. The US opposed this. For many years, the US had enough allies to block the change, while the ROC directly vetoed the Mongolian application.
When the European empires decolonised, many new states applied to join and the expectation was that most of them would be Western allies. The USSR vetoed some applications for UN membership, because it hoped to bargain over the Mongolia/PRC issue and because it thought it had nothing to lose by offending the African and Asian governments. Mongolia was finally admitted when the US leaned on the ROC. The PRC finally replaced the ROC in 1971, as a consquence of the US decision to switch recognition from Taipei to Peking.
A few of the vetoes relate to other legacies of the Second World War. The USSR was even more hostile to Spain, which had a Fascist government unti the 1970s, had provided moral support to the Axis during World War II, and whose dictator (General Franco) had killed many Soviet-backed Communists during the Spanish Civil War. Likewise, Ireland had been neutral during WW2 (Taoiseach de Valera infamously sent condolences on the death of Hitler) which the USSR was unhappy about. Again, this made much more sense if you see the UN as a potential military alliance.
Finally, Italy is a combination of both of the above. Both London and Moscow could not quickly forget that Italy had declared war and invaded them quite recently. The USSR additionally argued that the southern European state had to be dealt with in the same way as the former German partners, Hungary and Romania. Their applications had been blocked by the USA on the grounds that they were not the democratic states envisaged in wartime agreements (of course the wording had been vague, otherwise the Soviets would never have made them).
EDIT: I presume this is obvious, but the applications of (South) Vietnam and the Republic of Korea were vetoed because the USSR recognized North Vietnam and Democratic People's Republic of Korea. Under the Hallstein Doctrine, most states (other than the WW2 Big Four) had to recognize one and reject one of the Chinese, German, Korean, and Vietnamese pairs of adversaries.