Most people who've played in a role-playing game know that some people make poor DMs. A few people would rather simply tell a story than participate in one. There's a trick to this in that most people who DM with a heavy railroad could be a great dynamic and responsive DM if they could just ditch the rails. What are the warning signs of the train hitting the tracks? Here's a few: your DM never takes notes; player ingenuity is thwarted at every turn; the game displays little to no response to surprise actions from the players; NPCs are hogging the spotlight. Not all of these automatically tell you that there are rails but they are things worth examining.
A sign that the stars may be unalterably aligned against the party is when the DM simply doesn't take notes – either during the game or after it. Unless they are one of the rare few individuals with impeccable recall then they shouldn't rely on their memory entirely for such an endeavor. Hosting, calibrating and simulating a guild, town, nation, world or an entire universe are very large tasks. While it's perfectly acceptable and even common for them not to care what combination of equipment you've purchased from the blacksmith's stockpile, it's a bit unsatisfying if they keep the game running according to how they feel it should run (thus, rails) instead of as a collaborative effort. Lack of note-taking doesn't signify a railroaded campaign but it can be an indication if other signs are present.
Also along these lines are DMs who insist that whatever bad thing is happening is going to take place no matter what anyone in the party does. "I'm going to follow that man who just cursed at me out of the bar." This is a reasonable and I daresay predictable course of action. When the DM tells you that they are just too fast or they vanished almost immediately upon leaving the door then there's one of two options: either they really are that quick/skilled or they are under the "NPC protection program." In the former case it's a fair bet to say that some powerful or important NPCs definitely would have an impressive escape route at their disposal. If this happens all the time with every NPC and your party hasn't yet discovered how to perform such a disappearing act themselves after a good deal of research then it's a safer bet to anticipate rails. The same goes for things which simply happen despite the party preparing for them. When a barkeep that you've hidden in the cellar inside a wine barrel turns up missing anyway then it's tough luck. Should that happen every time the party tries to thwart the "grand plot" then either there's a leak in the party or in the DMs ability to host a collaborative story.
Both of the paragraphs preceding are rooted to the same cause. This is the same cause at the heart of every pure railroad game as well as every poorly run railroad plot. Such a cause is nothing other than a lack of free will. When engaged in a role-playing game there are certain expectations that are universal. The players will play their part as main characters in an interactive story and the DM will provide the parts of the story that the players are left out of (such as bartenders, blacksmiths and villains). Included in this and emphasized in this paragraph are the terms "interactive story" – people want it to look like their character's choices (and thus their own) are influencing the tale with which they are investing their time and energy into. Pure railroads are great for books because the reader passively follows along and has no ability to contribute. Interactive, by definition, isn't compatible with the idea of a pure railroad. Even strong railroads such as most video games are only so interactive. Tabletop role-playing games stand as the most interactive storytelling device in existence. Any DM who railroads so strongly they don't account for the other contributors to the story should be writing books and not hosting games.
Another piece which deserves strong emphasis is the players playing as main characters while the DM covers the parts where "the players are left out." A very good sign that there's a track in your future is when NPCs take up more game time than the players do. I'm not talking about a villain talking for a long time about their master plan or the king issuing orders to the party. The issue I'm directing attention toward is when two NPCs talk together for a while with the players having nothing to do with it. Players should be saving the day in most cases. If NPCs lead the story then you there are good odds that there's a track. Those NPCs are going to bring you to them regardless of your actions.
It can be fairly rough gaming with a DM who is using a heavy railroad. Do note that they are allowed to influence the story as well. Some DMs might not even realize that they are railroading the party. Regardless it falls on each person to ensure that the gaming experience is positive for everyone. Clearly I mean to say that if your DM is railroading you then being angry with them shouldn't be the first choice you make. Understanding is best promoted not by oppositional defiance but by a foundation of empathy. Besides, once those rails aren't so strict the person might make a great DM as well as gamer. Happy collaboration!