Many companies do levels for Software Engineer as SE1, SE2, SE3... etc.
For my title on my resume, should I list it as "Software Engineer 1/2/3" or is "Software Engineer" sufficient? Are there benefits of one over the other?
"Software Engineer 2" tells me nothing that "Software Engineer" doesn't, as I have no clue what "Software Engineer 2" means at your employer. At some places, it might mean "got through three months without being shown to be absolutely useless" while at others at might be something which is hard-earned after two years of high quality work.
Tell me what you did in your role, not what was written in some HR system somewhere.
Often no, but there are exceptions for certain industries and companies
Philip Kendall's answer covers most of it. I know a company where "Senior Software Engineer" is anyone full time or above and "Software Engineer" is an intern only title. There are people who have a year less experience than I (which is less than 2 years) do who have Senior in their title. At my prior company, software engineers had no levels. If you were a new grad, you were a Software Developer and if you had 15 years experience and were previously an Architect, you were also a Software Developer.
The exception is in industries that have very clearly defined levels such as banks. In a bank, Analyst is the lowest professional rank. Plenty of software engineers at banks are Software Development Analyst or Programmer Analyst. Associate comes next. Manager/Vice President is next (even if you don't manage anyone or lead anything). Then it is Director (again, even if you don't manage anyone). Then Managing Director (they usually have some reports, but again, not always). Throughout a bank, if you have an entry level professional job, there is a good chance "Analyst" is in the title.
Those ranks carry meaning as they are fairly standard within their industry, so if you are going from one bank to another or from a bank to an old stock insurance company, it will be clear to their hiring team what level you were at.
An exception should also be made for companies like Amazon. SDE II is widely known to be intermediate developer.
Whether a Software Engineer 2 is “above” a Software Engineer 1 is specific to individual companies. In some companies the numbers go up, in others they go down.
Including a prior (or current) job title in your resumé at all isn’t necessarily the right thing to do. Some people will recommend that you instead list your core responsibilities.
It’s more important to be clear what position you are applying for.
Just be aware that as with job titles, what works at one company may not work at another — there is no perfect resumé that works everywhere.
You should list the start and end date for each role, a description of your accomplishments and responsibilities in that role, and the relevant title. You should, of course, focus on how you brought value to the company/team, and not just enumerate the bullet points in the job description for the role. What a hiring manager wants to know is: "What skills were you demonstrating during this period? How could I leverage those skills on my team? Were you advancing? Did your management recognize your advancement with a promotion?" Across the software engineering industry, few titles will transfer directly, so the absolute value of a title is barely relevant, unless the hiring manager is familiar with the way your employer actually levels engineers. Even then, this knowledge becomes less useful the larger the company is, since different teams within the same corp can level their staff quite differently. Instead, the hiring manager should be looking to see a shift in responsibilities and accomplishments when you received a title change, and using that to calibrate their expectations of that title/role at that corp.
What you absolutely do not want to do is to label yourself as a Software Engineer 2/3/etc. That signals to hiring managers that you are clueless about the larger industry, that you don't talk to friends in other companies, that you haven't interviewed with anyone else who has a different levelling system, etc. In the worst case, you might sell yourself short by giving yourself a title below what a prospective employer is inclined to give you based on your work history and interview performance. Of course, they won't necessarily tell you that or give you the higher title anyway. Instead, they will say: "This person doesn't really know their market value, so we can safely under-level and underpay them. Score."
You want to make sure that your resume is compatible with the offers that you are applying to, regardless of the titles on offer. This works best when you imply with your resume that you understand the strong variance in levelling across companies, and that you will evaluate their position based on the expected responsibilities, and not just on the title/comp. If you have friends in other corps, you should talk to them about how levelling works at their company. It's especially good if you have a friend that works at the companies you are applying to. Getting the inside track on reasonable expectations for titles you could be hired at shows the hiring team that you do your homework, you don't have unrealistic or naive expectations, and that someone who already works for them might vouch for you. If you don't have any friends at other companies, then you really need to go to some industry networking events like user groups, OSS meetups, job fairs, etc. Most likely, you worked with some folks who moved on to other companies. It's good to stay in touch with some of these people for exactly these reasons. Keep your LinkedIn up to date to maintain your network.
In your overview, I would just use "software engineer" to describe you without any special qualifications (unless you have special qualifications, like "game dev", "data science", "front-end", "full-stack", etc. ...but remember that this will also pigeonhole you somewhat). If you have inclinations for management, you should consider adding a blurb about that in your objective statement as well. Your title is not your career. Titles are not like grades. They don't transfer well across companies. The hiring manager wants to see progression across time, but the actual titles achieved are less significant than your responsibilities and accomplishments.