Messages that look alarming and are not.
DEPRECATION NOTICES
A warning that something will be removed in future. Not a failure.
Useful before a PHP upgrade, and not a reason to act urgently.
NOTICES AND WARNINGS
Usually harmless. They should not be displayed on a live site, and their presence in a log is normal for many applications.
Turn off display, keep logging.
404s IN THE ACCESS LOG
Scanners probe every site for common vulnerable paths. A stream of 404s for files you never had is background noise, not an attack succeeding.
GREYLISTING DELAYS
A first message from a new correspondent arriving minutes late. Normal, and it stops after the first message.
CERTIFICATE WARNINGS ON THE SERVER HOSTNAME
Connecting to the server hostname rather than your domain may produce a name mismatch. Expected.
HIGH LOAD BRIEFLY
Short spikes are normal. Sustained high load is the concern.
WHAT IS ACTUALLY A PROBLEM
Anything preventing visitors using the site, anything preventing mail arriving, and anything indicating a compromise.
Judge by effect rather than by how alarming the message sounds.