Allow me to give more detail on them, based on Coldreactive's comments.
Hearing electric crackling, followed by a heartbeat, usually denotes a Vizbolt E-trial that can be triggered somewhere on the map. Sometimes, it's a Judgement E-trial (Disabling bombs).
The normal buzzing followed by a heartbeat denotes a Baize E-trial that could happen in the map, somewhere.
Lillipan Screams are usually the Rescue/Gesture E-trials, or in the abandoned mine, the E-trial where you have to protect villages.
The whooshing effet that is accompanied by an effect along the bobrder of your scream indicates a boss that doesn't have a distinct cry.
HEARING a cry denotes a boss, too, but at least you know what boss it is. Additionally, on the floating continent and in the city, if you hear the Quartz Dragon's cry, you usuallyt get a chance to use your Action button to view it flying around in a circle, off in the distance.
A crystal-sounding effect along with prismatic ripples indicates a crystal Defense E-trial. THESE ARE VERY IMPORTANT TO KNOW. They are difficult, as the crystals don't have much health, and tend to spawn about 5-6 of them throughout the E-trial. If two break, you fail. But if you complete it, PSE Triggers go through the roof, and if you're in a large party, a 4-way Cross Burst with at LEAST Two WAN MORs is guaranteed.
There's also a variation of the first one, along with the sound effect that sounds similar to a Gilnach using its heal ability right at the end. This indicates a Clone E-trial, nearby. They're very rare, and only happen in Single-party areas.