When surfing the Web on a smartphone, most of us stick with the browser that came with our handset. That experience can be clunky, though, and a slew of mobile browsers are trying to break into a market dominated by Apple and Google.

It’s a struggle reminiscent of the “browser wars” of the ’90s, in which Netscape Navigator and Internet Explorer fought for dominance on desktop computers—and also of the more recent battle for market share among IE, Firefox, and Google’s Chrome browser on desktops and laptops.

Mobile browsers like Opera, Rockmelt, Dolphin, and the brand new Futureful are sparring with the built-in browsers on the iPhone, iPad, and Android-running smartphones and tablets, hoping to grab a percentage of the growing market for surfing the Web on these smaller screens.

The odds of success appear slim for most contenders. According to data from Net Applications’ NetMarketShare, Apple’s Safari browser captured nearly 61 percent of the mobile browser market in January, while Google’s Android browser had more than 21 percent. (Google recently started including its Chrome browser on high-end Android devices, but Chrome’s market percentage is still tiny, according to NetMarketShare.) Opera Software’s Opera Mini browser came in third with about 10 percent of the market.

But as the smartphone and tablet markets expand and people shift much of their online time from laptops and desktops to mobile devices, mobile browser makers see an irresistible opportunity to innovate and—just maybe—become the preferred pocket-sized gateway to the Web.

Mobile-browser companies are fond of saying that the market leaders have mainly just taken a desktop Web browser and plopped it onto a smaller screen, creating an experience that isn’t great for smartphones in particular and doesn’t take advantage of mobile devices’ touch screens.

“It’s really frustrating,” says Eric Vishria, cofounder and CEO of mobile browser Rockmelt. “They really haven’t pushed the limits given how important this software is.”

Rockmelt’s free iPhone and iPad app tries to make browsing faster and more convenient by opening up with a full screen filled with squares of content determined by your interests, which it gleans if you allow it to connect to your Twitter and Facebook profiles. You can swipe interesting-looking stories to the right to save them for later (you can read these offline, too), or swipe to the left to indicate that you don’t like a certain type of story. You can post reactions to Web pages by tapping tags like “want” or “WTF” and follow friends and websites within the browser to see what they’re reacting to. Rockmelt shows items you click on in your content feed in a simplified view that includes, generally, just one large image and text; to see the content as it appears at its source, you tap a little sunglasses icon in the URL bar.

Rockmelt also attempts to overcome network latency issues by preloading content on its servers and piping it down to users. Vishria declined to give exact figures but says that Rockmelt has hundreds of thousands of users (there are also 4.3 million users of its discontinued desktop browser).

And Rockmelt has a cheerleader that any browser company would envy: Marc Andreessen, cofounder of Netscape, who sits on its board. The venture capital firm he cofounded, Andreessen Horowitz, is one of the company’s investors.

This kind of support signals an opportunity for mobile browsers to succeed, says Jason Davis, an associate professor at the MIT Sloan School of Management who studies innovation and entrepreneurship.

MoboTap, the company that makes the free Dolphin Browser, is betting on that opportunity. Dolphin, for the iPhone, iPad, and Android smartphones and tablets, aims to stand out with mobile-friendly features like the ability to set up unique gestures that load specific websites. You do this by tapping a little dolphin icon at the bottom of the browser screen and drawing your symbol—I made a “J” to take me to the blog Jezebel. Users can also pay $1 for Dolphin Sonar, which lets you shake your phone to activate voice controls for searching, browsing, opening new tabs, and sharing Web pages.

Dolphin’s products seem to be doing well. Its apps have been downloaded more than 50 million times, and the browser has about 17 million active users, says Edith Yeung, who heads up Dolphin’s marketing. She also notes that the browser is being preloaded on smartphones in Japan and China through partnerships with wireless carriers and device makers, and that Dolphin is talking to others who are interested in making it the default browser on phones.

Still, for any mobile browser, just getting discovered is an issue. Apple’s and Google’s application stores are extremely crowded, and many consumers wouldn’t even think of going beyond the built-in browser on their device. Rockmelt’s and Dolphin’s user figures are tiny in comparison with the millions of iPhones and Android smartphones that are snapped up each quarter, every one of which comes preloaded with a browser. In the fourth quarter of 2012 alone, nearly 200 million of the 217 million smartphones shipped worldwide ran on Apple’s iOS or Google’s Android platform, according to Strategy Analytics.

“Most people are probably not even aware they can download a different browser,” says Andrei Hagiu, an associate professor at Harvard Business School who teaches a course on the Internet browser wars.

This isn’t keeping new mobile browsers from coming, though. A new entrant to the field is Finnish startup Futureful, whose free iPad browser is focused on predicting what you’ll want to check out online and bringing it to you rather than making you go find it yourself.

Like Dolphin and Rockmelt, Futureful emphasizes browsing without typing, though it, too, will let you type in a URL if you really want to. Users sign in with their Facebook or Twitter account, and Futureful builds an ever-changing profile based on your interests. The browser uses this information, along with data about what similar users like, to recommend content in little tags that float across the top of the screen. As you use it over time, it will get better at determining what you want to see, says cofounder Jarno Koponen, and users can also mark items they like to help Futureful learn faster.

Like Rockmelt, Futureful has a big-name investor: Janus Friis, cofounder of Internet telephony company Skype.

Could too many alternative mobile browsers be a bad thing? Hagiu thinks so if each browser requires website makers to optimize their sites differently. On the other hand, if mobile browser companies adhere to a common set of Web standards, such as HTML5, this is unlikely to be an issue.

Right now, four out of every five minutes devoted to media consumption on a smartphone are spent using apps, according to data from comScore, while just one minute is consumed by mobile Web surfing.

Eventually, the proportion is likely to change as websites are revamped for the smaller touch screens and as mobile gadgets become more powerful computers. For now, though, these minority mobile browsers have some time to experiment.