A lot of people on here will likely disagree with me and this opinion.
In my opinion the answer lies in your own motives and intentions.
If you'd like to be more cognizant of the market, I'd just dive in and buy a few companies you like. Many people will say you shouldn't pick your own stocks, you should buy an index fund, or this ETF or this much bonds, etc. You already have retirement savings, capital allocation is important there. You're talking about an account total around 10% of your annual salary, and assuming you have sufficient liquid emergency funds; there's a lot of non-monetary benefit to being more aware of the economy and the stock market. But if you find the house you're going to buy, you may have to liquidate this account at a time that's not ideal, possibly at a loss.
If all you're after is a greater return on your savings than the paltry 0.05% (or whatever) the big deposit banks are paying, then a high yield savings account is the way I'd go, or a CD ladder. Yes, the market generally goes up but it doesn't ALWAYS go up. Get your money somewhere that it's inured and you can be certain how much you'll have tomorrow.
Assuming a gain, the gain you'll see will PALE in comparison to the deposits you'll make. Deposits grow accounts. Consider these scenarios if you allocate $1,000 per month to this account.
1) Assuming an investment return of 5% you're talking about $330 return in the first year (not counting commissions or possible losses).
2) Assuming a high yield savings account at 1.25% you're talking about $80 in the first year.
Also remember, both of these amounts would be taxable.
I'll admit in the event of 5% return you'll have about four times the gain but you're talking about a difference of ~$250 on $12,000. Over three to five years the most significant contributor to the account, by far, will be your deposits.
Anyway, as I'm sure you know this is not investment advice and you may lose money etc.