Regarding China, there are three aspects that have been used for most of the Communist Party's reign. I'll also talk about the Cultural Revolution at the end, but that was just one short part of the Party's history.

First, is Communist Party membership. Since the party took over, Communist Party membership has of course been required for work as a politician, but it has also been necessary for civil service jobs (including things like being the head of a school or hospital), and strongly beneficial for other jobs. For example, it's not required to be a businessman, but having membership allowed businessmen to flourish through links to local officials. Before Deng Xiaoping's reforms, it was even more important, as all industry was under direct government control. Communist party membership was required to be the leader of a work cadre, for example. Now, while as I understand it, religion is not explicitly banned, party members have to sign up to the Party's statement of beliefs, which includes atheism. Thus, most religious people will consider joining the party to be in conflict with their religion. They are thus denied a leading role in society, politics, and prior to Deng Xiaoping's reforms, business. Some religious people (especially Buddhists) do join, as it's easy to hide, but generally stay at the lower levels. There are plenty of ethnic Muslim members, for example, who don't eat pork (which is presented as a cultural practice), but going to Mecca would cause them to lose membership.

Secondly, is regulatory control of religion. In the 50s, a number of organisations were set up to regulate religious activity by Zhou Enlai, and these became compulsory. These ensure that religious institutions are unable to gain widespread influence or power. Note that the purpose of this was primarily political, not anti-religious, but it had an influence on weakening religion. It did also, heavily cripple those religious groups who didn't fall into the limits of the 5 religious organisations, for example smaller Christian denominations, which are completely illegal. This led to the growth of the House Church movement. Interestingly, while one of the stated purposes of these regulations are to stop the formation of cults, China actually developed a reasonably serious problem with Christian-based cults which had its roots in the secretive peasant churches of the 70s and 80s, who were unable to connect with mainstream Christianity.

Third, are laws restricting religious activities. Missionary activity by outsiders was made illegal (this, it should be said, has roots in the 19th century, when the Qing tried to restrict Missionary activity and were forced to lift restrictions at gunpoint by western powers) in the early years of the Communist regime, as was evangelism. Exactly how evangelism has been defined is somewhat nebulous, the same as political protest, so has generally depended on the local officials and police. This restricts the growth of religion. There have also been restrictions on young people being allowed to attend Church or Mosques. It should be said that the restrictions on religions have generally been most strict on, in the following order, Tibetan Buddhism, Islam in Xinjiang (but not the rest of China) and Christianity, in roughly that order. This is because these were the religions that were considered most threatening to the regime politically. Taoism and Chinese mainstream Buddhism, while still restricted, have not, for the most part, been as persecuted.

Another thing to note is that religious restrictions are less strict in the modern day than they were before Deng Xiaoping, as the Constitution guarentees freedom of religion (though it would not be considered free by western standards).

Now, the absolute worst time for religion was the Cultural Revolution. This was a period in the late 60s/early 70s where Mao, worried that China was backsliding towards Capitalism and Feudalism, encouraged teenagers to fight to ensure the revolution, against the "4 olds: old ideas, culture, customs and habits". They formed the Red Guards, and persecuted officials, teachers, intellectuals, and also religion. Depending on the local Red Guards, priests, monks, nuns and Imams might be subject to anything from public humiliation to being beaten to death. Many clergy were sent to camps, Monks forced to demolish their own temples at gun-point, and Qurans burnt. Confucius' tomb was vandalised. In Tibet, all but 8 of the over 6000 monasteries were destroyed, and by the end of the cultural revolution most Tibetan monks and nuns were dead or in prison. 1600 Hui (a Muslim ethno-religious group) were massacred in Yunnan. You can see that this was not strictly speaking an anti-religious crusade, it was a crusade against Class enemies, of which religion was considered to be a tool. You can also see that ethnic minorities were unduly effected. After the Cultural Revolution finished, direct religious persecution was stopped, but the pre-Cultural Revolution restrictions (as I noted above) were maintained.

If you'd like to know more about modern religious persecution in China, I'd look at Falun Gong, a religion suppressed in the late 90s, and if you want something current, the recent cross-removals in Wenzhou (it's not an extremely serious case of religious persecution, but it's an example of normal religious persection). This subreddit has a rule about not discussing the last 20 years though, so I'll leave that for you to read.

I don't know much about the Soviet methodology (well, actually I know a guy who was purged because of it, but that would come under personal anecdotes), but if you're interested I could talk a bit about how Religion has been and is suppressed in China? It's a Communist country, but not Soviet, so I'm not sure if that's what you're looking at.