#2 in @Scott Trench post - He is referring to the IRS tax code that allows you to get a $250k exemption (if single) or $500k exemption (if married and filing jointly) on your capital gain of the sale of your primary residence. To qualify, you must have lived in the home as your primary residence for 2 of the previous 5 years. THis is good if you intend to sell and buy a new primary.
That said, unless you run the numbers for each option and lay them out side by side, there is really no way (other than guessing) which option is better, sell primary, rent primary, cash out refi primary, etc. It would be best to run a financial analysis with your CPA on each option and then each option that delivers a bucket of cash to you, how do you invest that - there will be choices to make there as well which means more options to run financial projections for each and compare those side by side.
On the negative side, moving SUCKS most of the time so there is that. Your family (kids schools, friends, your friends, etc) should all come into play in your decision as well so if staying is "best for personal life for you and fam, then pick that and use your other options to see where and how to invest.
Here are a few options to consider along with what you have already named: Investing in a syndication with your savings, cash out refi, or any other liquid funds you have can be a goo place to earn double digit annual returns and get economies of scale with your investments. Managing singles, duplexes and small multi's can be a difficult job, especially if you do not have experience or know how which means you are subing that out to a property manager in which case you had better get a good one if you care anything about your money. Having large multi's provides better economies of scale with professional managers on site, systems in place, etc. This can all be learned by investing with and watching a professional syndiactor/sponsor along the way and then perhaps doing one yourself as the operator. Option 2 is investing capital with other partners with experience. A good partner can catapult your investments and a bad one can sink the ship so use caution here.
Pay close attention to probabilities of future rents for your primary if you choose that option and what the future appreciation growth could be (don't get crazy here, use conservative numbers). If renting the primary delivers you less cash flow but a much higher appreciation gain over time compared to selling for a higher cash flow position but lower appreciation gain, then you have to decide with your CPA which fits your needs better. Some need cash flow to live while others focus more the growth (appreciation). As an example, lots of people trash talk CA and say we have no cash flow. While that is not a true statement, it is fairly accurate as cash flows are minimal here, but appreciation kicks rear ends. If you compared 2 properties, 1 from CA and 1 from TX, the TX property over last 5 years will likely show positive cash flow and some appreciation (the best of both worlds scenario) and the CA property would likely show break even to cash flow loss, but hundreds of thousands in appreciation. At the end of the day after 5 years after adding up cash flows (or losses) with appreciation gains and liquidation costs, the CA property would win out each time by a landslide. Granted we are talking about the last 5 years which had unprecedented appreciation. Hopefully you get the picture from my long winded post here.
In summary, get with your CPA and forget about the rest of our opinions, use data and facts to compare each option side by side, then apply your family situation into the decision process and your decision will be much more clear!