Holland America's Dress Code doesn't anywhere stipulate that jeans are inappropriate for Smart Casual Evenings. True, it used to be stated in the Dress Code that jeans were not acceptable in the evening hours on Casual nights, but that is no longer the case. Hence, on Smart Casual Evenings if you would like to dine in the main dining room there really is no reason not to do so. If your jeans aren't worn out or tattered, with holes, and if your shirts are not t-shirts but, rather, polos or nice print shirts with a collar, you shouldn't have any problem.
As for Formal Nights, you can certainly dine in the Lido in jeans and sport shirt/polo shirt. The Lido is always Casual.
IF you don't want to carry formal gear, but would be interested in actually engaging in the Formal Night "festivities," you might want to consider going the Formal Wear rental route. It's an excellent service, not expensive, and I now a bunch of people who rave about how much nicer it is to not have to carry formal gear with them. You'll find information about Cruiseline Formal wear on the HAL website, or HERE
For reference purposes, here's HAL's evening Dress Code:
Evening dress falls into two distinct categories: Formal or Smart Casual. Smart Casual can be defined as slacks and collared shirts for men and casual dresses, slacks and informal evening wear for women. T-shirts, swimsuits, tank tops and shorts are not allowed in the restaurants or public areas during the evening hours. On festive Formal evenings, ladies usually wear a suit, cocktail dress or gown and gentlemen wear a jacket and tie, dark suit or tuxedo. There are approximately two formal nights per week.
In order to complement your fellow guests, Holland America Line asks that you observe the suggested dress code throughout the entire evening.
Formalwear for ladies and gentlemen can be pre-ordered for your use during your cruise by calling Cruiseline Formalwear at 800-551-5091 or make your reservations online. Your formalwear will be in your stateroom when you board.