Please note: All of the below is based on American English
Great question!
Grammatically speaking, a subject is always required. However, many of us choose to write grammatically incorrect sentences on purpose. This takes place in chats (verbal or text), but will not take place in proper text such as newspapers or books.
( 1 ) Subject = I : When we leave out the subject "I" the purpose is to sound less arrogant. If we talk about ourselves a lot, it will sound self-centered. You'll often see "I" omitted from Facebook posts. For example, you may see a Facebook post like:
Went to the grocery store today to pick up some apples. Couldn't believe what I saw. There were no apples in the whole store!!
This does make the sentence less formal. I would not omit "I" if I was speaking to my boss. However, in most conversations this is perfectly acceptable.
( 2 ) Subject = inanimate object : There is no particular reason why we leave out the subject in this case. It also does not appear to make the sentence significantly less polite. I would have no problem having the following conversation with my boss:
Boss: "How are the reports coming along?"
Me: "Almost finished! Should be done before the end of today."
It's fine. Some linguists refer to this kind of omission as a conversational deletion.
Conversational deletion is the tendency to omit portions of a sentence, usually the subject, when those portions refer to people, places, or things deemed obvious or implied in the conversation. —tag info for conversational deletion.
The conversational deletion is very common and perfectly idiomatic in speech. It's not likely to run afoul of any rule in informal situations.
Best Answer
In certain types of writing, for example:
... we often don't have a lot of space or a lot of time. It is common in these types of writing to drop the Subject - and the auxiliary verb too - if the reader can easily understand who the Subject is. Very often, but not always, the Subject is the speaker. Usually we understand from the context.
These sentences mean:
Notice that in the second example, the Subject is you, not I. Also notice that the auxiliaries be and have are missing from some of these examples too.
This is sometimes called diary drop by linguists.