This is mainly for people who are competitive powerlifters but I figure others could use this advice as well. Maxing/singles is a skill you need to learn. There is a mental preparation and a physical preparation that needs to be developed. Physically, one needs to develop inter and intra-muscular coordination. In massive layman's terms, your body needs to learn how to turn EVERYTHING "on" for that one max attempt. No wasted energy; everything directed into the bar that is moved. Mentally it helps to be insanely competitive. The only way to get good at competing is...competing. And I'm not just talking about doing powerlifting meets; any kind of sport/activity that requires you to get out of your mental comfort zone and have some consequences will help. I know both of these can be developed - and I know that these are also inherited. As always, some genetics will always come into play.
Physically, there are three ways of developing your ability to perform singles.
1. Train for a long time and build a huge base of strength and muscle. This is not a secret or a surprise to anyone. This takes time and then some more time.
2. Train main lifts explosively. Every rep needs to be controlled eccentric and explosive concentric. Treat every TRAINING/WORK rep like you are maxing out. Obviously, light warm-up sets can be done differently but don't just wimp these weights up, either. Be under control. But I see way too many people squatting a 10 rep PR set like they are saving something in the first few reps. Snap those weights. On the flip side, don't do the reps sloppy. Being explosive means control.
3. AFTER the main work, usually when you are feeling good, take a single. But here's the rule: NEVER MISS. Feel free to take 1-3 singles after your main work and be sure to treat these with the respect and power they deserve. Do not over do this. You don't need to do these every single week. If you don't know how to do this, switch your training to 3/5/1 - and take a single after the 3x3 week and the 5/3/1 week. Also, don't sacrifice your work sets to "save" something for your single. it's training, not testing. You are better off doing the single after a hard work set; it will force you to mentally and physically prepare. Mentally work through the single prior to attempting it. Prepare your mind for what is about to come. Visualize every aspect. Once again, never miss the single. Choose wisely.
Personally, I never had a problem with singles (once I developed a base of strength/size) because I always treated every rep like it was 500lbs. Never wimp the weights off the rack/floor. I always tell myself that regardless of what I'm doing, the first rep is most important. it sets the tone, physically and mentally, for every rep you do after. Make that first rep absolutely killer. Prior to taking the bar off, I feel the knurling in my hand and dig in. I close my eyes and see what is going to happen. I can feel the weight in my hands and on my back - it feels light because I choose for it to be light. I am controlled in my breathing and in my approach - when the bar is descending, I can feel all the energy build up in my body. And when the time comes, my body is ready to snap the barbell in half. There is no half-ass approach at the bottom of the lift.
This mental approach is done for every single work set with a barbell. If you want to get good at singles, take some time to build your body. But you best have your mind ready to. Don't wait until the meet to practice. You have thousands of reps to practice; use them.