TLDR
Keep using 0.y.z while in development phase (and your API is not in prod), worry about it when in Production
Long Story
If you DO want to follow semver:
Btw, the answer is based on https://semver.org/:
SemVer 2.0 is designed as set of rules to clearly communicate the nature of the changes contained in your new release.
X.Y.Z
X - changes here represent no backward compatibility
Y - backward compatible new features
Z - backward compatible bugfixes
But it also says on item 4:
"4. Major version zero (0.y.z) is for initial development. Anything MAY change at any time. The public API SHOULD NOT be considered stable."
You may mark your version with a pre-release tag after the version, as suggested by Rohit, (you may, but you don't have to). The full spec could be:
<version core>"-"<pre-release>"+"<build>
Where:
<version core>: X.Y.Z (as stated before)
<pre-release>: your pre-release tag (check semver site for details on what it should contain)
<build>: build identifier
The spec also gives you some hints on how to deal with versioning during development phase in the FAQ:
How should I deal with revisions in the 0.y.z initial development phase?
The simplest thing to do is start your initial development release at 0.1.0 and then increment the minor version for each subsequent release.
"How do I know when to release 1.0.0?
If your software is being used
in production, it should probably already be 1.0.0. If you have a
stable API on which users have come to depend, you should be 1.0.0. If
you’re worrying a lot about backwards compatibility, you should
probably already be 1.0.0."
Doesn’t this discourage rapid development and fast iteration?
Major
version zero is all about rapid development. If you’re changing the
API every day you should either still be in version 0.y.z or on a
separate development branch working on the next major version.
This basically says that you can stay on 0.y.z if you don't have heavy dependency on your API. If your clients treat your API as their "production" or "production ready", maybe you should move to 1.0.0 and follow the rules.
The moment you go to production you should be already on 1.0.0