Note: You can only measure sleep accurately with a sleep study (polysomnogram) done in a lab which measures things like brain waves [electroencephalogram (EEG)], eye movement [Electrooculogram (EOG)], muscle movement [Electromyography (EMG). ]
See: Sleep Study | Johns Hopkins

Sleep-tracking devices: Dos and don'ts | Mayo Clinic
"Consumer sleep trackers are pretty accurate at tracking total sleep time."
A sleep tracker can make an educated guess about your sleep stages.
But the only way to accurately identify what stage of sleep you are in is with clinical sleep study (polysomnography).

Reviews:

iOS
Note: Apple does not have their own sleep app.
 Apple is rumored to be releasing their own sleep app
 in a new release of watchOS at the end of 2020.

There are 2 types:
1. Detect sleep thru sound an motion sensing with your phone on the bed.
2. Use an apple watch

1. Without watch:

The Best Sleep-Tracking App for 2020 | Reviews by Wirecutter
As a health journalist, I've followed the scientific research on sleep
 for more than a decade. 
 For this guide, I interviewed product specialists
  and brand executives in the sleep-tracking field, sleep doctors 

SleepScore 4.3 (4.6K)  premium $49/yr
It's more intuitive and provides deeper sleep analysis
 and advice than other apps.
 SleepScore offers about as much information as Pillow and more than Sleep Cycle.
 It's more transparent about the science behind its app.

Tom's Guide


Sleep Cycle 4.7 (175K) $26/yr American Sleep Association
Tom's Guide
Uses sleep cycle theory to wake you at just the right time,
 ensurin you're getting optimal rest. 
Simpler interface, less data.
A good choice for people who simply want to see sleep patterns
 and how they correlate with diet and exercise.



2. Apple Watch
The best sleep tracking apps to download for your Apple Watch | Wearable

AutoSleep Track 4.7 (50K) $2.99
When put up against a dedicated sleep tracker,
 it produced close results,
 but you can tweak the sensitivity for better accuracy.

Sleep++ 4.2 (4.8K)
These are the best applications for sleep tracking with Apple Watch | 9to5mac

AutoSLeep 4.7
Pillow Automatic Sleep Tracker 4.4 (76K)
Best For Figuring Out Why You Can't Sleep  WomensHealth

Sleepzy 4.2 (9.6K)
Sleep Watch 4.7 (178K)

Best Sleep Apps For Apple Watch To Try in 2019 | TechWiser
Pillow Automatic Sleep Tracker 4.4 (42K)
Sleep Watch 4.6 (89K)
Sleep++ 4.3 (4K)

Americans spend an average of 7 hours and 18 minutes in bed each night. They go to bed at 11:39 p.m., wake up at 7:09 a.m., spend 23.95 minutes snoring, have an average sleep quality of 74.2 percent, and rate their wake-up mood at 57 on a scale of 100. Feb., 2019
Source: What we know about Americas healthiest, happiest and best-rested people - Sleep Cycle

What Is A Sleep Score and How Is it Calculated? | SleepScore Labs
SleepScore measures your sleep on a variety of levels. It monitors and analyzes your total sleep duration, your time taken to fall asleep, your time in light sleep, deep sleep, and REM sleep, and the number and duration of awakenings you have each night.
These variables are then measured against known sleep averages for your age and gender, which have been established by well-published research, and expert advisors. See Sleep Quality under health.
What Is A Sleep Score and How Is it Calculated? | SleepScore Labs

optimal values for sleep each stage are 5% Awake, 25% REM, 35% deep, and 35% percent light sleep.
Source: How does pillow calculate sleep quality? Pillow

Sleep quality ranges from 25% - bad to 90% - great

Generally accepted optimal values for sleep each stage are 5% Awake, 25% REM, 35% deep, and 35% percent light sleep for an adult,

It uses a patented technology.

            Pillow                                          Sleep Cycle

Review of Sleep Cycle App for Timing Awakenings | VeryWellHealth
Some of their comments:
There are many graphs, tracking sleep quality, Heart rate, Sleep time, all the way up to sleep quality affected by the moon.
Some Issues:
If you have a soft mattress pad or other bedding, the accuracy will suffer even more because movements will be missed.

It's called Sleep Cycle Alarm Clock because it has a feature where you set the time you want to wake-up and 30 minutes before that it monitors signals from your body to wake you softly when you are in the lightest possible sleep stage. My and others experience was that it woke us too soon.