Many big-name apologists attempt to demonstrate the historical accuracy of the New Testament (up to the point of claiming that every single thing described there is a completely accurate depiction of real events) by using techniques that might superficially resemble academically sound approaches, but are often outright ridiculous.
For example, miles and miles of text has been written trying to demonstrate how the New Testament writings that we have today are exactly the same as the original ones that were written. While there are very good reasons to believe that at least some parts of the New Testament, especially the Gospels, have been added later, this is nevertheless a completely inconsequential argument. It doesn't matter if the text is the unmodified original one. That doesn't give any credence to the veracity of what the text is actually saying. This is like arguing that Homer's Iliad depicts true events if we can demonstrate that the text we have today is the same as the original.
Another argument that these apologists try to hammer in is the claim that the text was written either by direct eyewitnesses or by people who had contact with direct eyewitnesses. No serious historian scholar today, Christian or secular, has any doubt whatsoever that the Gospels were not written by their alleged authors. There's no doubt that they were written by unknown authors at least half a century after the alleged events, and that they were most certainly not any kind of direct eyewitnesses of any such alleged events. There's also no doubt that the books of Luke and Matthew were inspired by, and variations of the book of Mark. Even many Christian scholars acknowledge this (even some of those who try adamantly to prove the veracity of the text as historical fact.) Nevertheless, this is also an inconsequential argument: Even if the Gospels had been written by eyewitnesses of a person who was the basis of the character of Jesus, there's absolutely nothing that would have stopped them from adding their own embellishments. (There are many known instances of exactly this kind of thing from even recent history, so it's in no way an inconceivable thing.)
One of the most ridiculous arguments that is presented in complete seriousness is the alleged vast amount of eyewitnesses to all of the events depicted in the Gospels and the other books of the New Testament. This is a ridiculous argument because the only source we have for the existence of these eyewitnesses is the New Testament itself. In other words, this is a circular argument: Trying to prove the New Testament as accurate using the New Testament itself. Trying to claim that it's accurate because of the vast amount of eyewitnesses, and the existence of said eyewitnesses is true because the New Testament says they existed. (And this isn't even going into the fact that even today it's easy to find thousands of alleged "eyewitnesses" to events that have demonstrably not happened. Group psychology can be a very strong phenomenon.)
One of the most fallacious arguments is trying to prove the historical accuracy of the New Testament by pointing out that some of its depictions are known to have happened with almost complete certainty (such as certain rulers having really existed.) This commits the fallacy of thinking that if some events are depicted accurately, that gives credence to the claim that all of the events described are true also. This is a ridiculous idea. It's like saying that since Homer's Iliad describes real cities, all of its events are therefore historically accurate.
The major problem with that is that apologists spend tons and tons of resources to try to prove mundane details about the Gospels as accurate, when those are mostly inconsequential and are not the things that skeptics have problems with. Whether this or that city really existed, or whether this or that person really existed doesn't really matter. Those things in no way attest to the veracity of the extraordinary claims made in the text. It's perfectly possible to write a fictional narrative to happen in real settings. We do this all the time.
Extra-biblical sources are also always brought up. This is highly deceptive because every single extra-biblical source that we know exists only mentions the existence of Christians, and none of them describe the events depicted in the Gospels. Nobody doubts that Christians existed in the first century; that's not the question. These extra-biblical sources are a very good argument for this. However, apologists are trying to go much further than that.
Perhaps the most ridiculous argument that apologists bring up completely seriously is the "empty tomb" argument. They say that because Jesus' tomb was demonstrably empty, that proves that Jesus rose from the dead. Yet the only source we have that such a tomb even existed is the very document that they are trying to prove as true. Again, this is a completely circular argument.
Apologists will also spent miles and miles of text to fight against alternative explanations for the empty tomb, as if that had any relevance whatsoever. They can't even establish in any credible way that there was any such tomb to begin with. (They will once again resort to the eyewitnesses argument, which is once again a circular argument.)