https://docs.npmjs.com/misc/semver

gistfile1.txt

Caret ranges are ideal when an author may make breaking changes
between `0.2.4` and `0.3.0` releases, which is a common practice.
However, it presumes that there will *not* be breaking changes between
`0.2.4` and `0.2.5`.  It allows for changes that are presumed to be
additive (but non-breaking), according to commonly observed practices.
 
* `^1.2.3` := `>=1.2.3 <2.0.0`
* `^0.2.3` := `>=0.2.3 <0.3.0`
* `^0.0.3` := `>=0.0.3 <0.0.4`
* `^1.2.3-beta.2` := `>=1.2.3-beta.2 <2.0.0` Note that prereleases in
  the `1.2.3` version will be allowed, if they are greater than or
  equal to `beta.2`.  So, `1.2.3-beta.4` would be allowed, but
  `1.2.4-beta.2` would not, because it is a prerelease of a
  different `[major, minor, patch]` tuple.
* `^0.0.3-beta` := `>=0.0.3-beta <0.0.4`  Note that prereleases in the
  `0.0.3` version *only* will be allowed, if they are greater than or
  equal to `beta`.  So, `0.0.3-pr.2` would be allowed.
 
When parsing caret ranges, a missing `patch` value desugars to the
number `0`, but will allow flexibility within that value, even if the
major and minor versions are both `0`.
 
* `^1.2.x` := `>=1.2.0 <2.0.0`
* `^0.0.x` := `>=0.0.0 <0.1.0`
* `^0.0` := `>=0.0.0 <0.1.0`
 
A missing `minor` and `patch` values will desugar to zero, but also
allow flexibility within those values, even if the major version is
zero.
 
* `^1.x` := `>=1.0.0 <2.0.0`
* `^0.x` := `>=0.0.0 <1.0.0`

gistfile2.txt

#### Tilde Ranges `~1.2.3` `~1.2` `~1`
 
Allows patch-level changes if a minor version is specified on the
comparator.  Allows minor-level changes if not.
 
* `~1.2.3` := `>=1.2.3 <1.(2+1).0` := `>=1.2.3 <1.3.0`
* `~1.2` := `>=1.2.0 <1.(2+1).0` := `>=1.2.0 <1.3.0` (Same as `1.2.x`)
* `~1` := `>=1.0.0 <(1+1).0.0` := `>=1.0.0 <2.0.0` (Same as `1.x`)
* `~0.2.3` := `>=0.2.3 <0.(2+1).0` := `>=0.2.3 <0.3.0`
* `~0.2` := `>=0.2.0 <0.(2+1).0` := `>=0.2.0 <0.3.0` (Same as `0.2.x`)
* `~0` := `>=0.0.0 <(0+1).0.0` := `>=0.0.0 <1.0.0` (Same as `0.x`)
* `~1.2.3-beta.2` := `>=1.2.3-beta.2 <1.3.0` Note that prereleases in
  the `1.2.3` version will be allowed, if they are greater than or
  equal to `beta.2`.  So, `1.2.3-beta.4` would be allowed, but
  `1.2.4-beta.2` would not, because it is a prerelease of a
  different `[major, minor, patch]` tuple.